Citation
Dynamic framework for subdivision surfaces

Material Information

Title:
Dynamic framework for subdivision surfaces
Creator:
Mandal, Chhandomay ( Dissertant )
Vemuri, Baba C. ( Thesis advisor )
Qin, Hong ( Thesis advisor )
Sahni, Sartaj ( Reviewer )
Fishwick, Paul A. ( Reviewer )
Cenzer, Douglas A. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
Department of Computer and Information Science and Engineering, University of Florida
Publication Date:
Copyright Date:
1998
Language:
English
Physical Description:
xv, 168 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Butterflies ( jstor )
Damping ( jstor )
Data ranges ( jstor )
Datasets ( jstor )
Degrees of freedom ( jstor )
Dynamic modeling ( jstor )
Geometric shapes ( jstor )
Matrices ( jstor )
Topology ( jstor )
Vertices ( jstor )
Computer and Information Science and Engineering thesis, Ph.D
Computer graphics -- Mathematical models ( lcsh )
Computer vision -- Mathematical models ( lcsh )
Dissertations, Academic -- UF -- Computer and Information Science and Engineering
Image processing -- Digital techniques ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )

Notes

Abstract:
Subdivision surfaces are extensively used to model smooth shapes of arbitrary topology. Recursive subdivision on a user-defined initial control mesh generates a visually pleasing smooth surface in the limit. However, users have to carefully select the intiail mesh and.or manipulate the control vertex positions at different levels of subdivision hierarchy to satisfy the functional and aesthetic requirements in the smooth limit surface. This modeling drawback results fromt eh lack of direct manipulation tools for the limit surface. In this dissertation, techniques from physics-based modeling are integrated with geometric subdivision methodology, and a dynamic framework is presented for direct manipulation of the smooth limit surface generated by the subdivision schemes using physics based “force” tools. In a typical subdivision scheme, the user starts with an initial control mesh which is refined recursively using a fixed set of subdivision rules, and a smooth surface is produced in the limit. Most often this limit surface does not have an analytic expression, and hence poses challenging problems in incorporating mass and damping distribution functions, internal deformation energy, forces, and other physical quantities required to develop a physics-based subdivision surface model. In this dissertation, local parameterization techniques suitable for embedding the geometric subdivision surface model in a physics-based modeling framework have been developed. Specific local parameterization techniques have been fully developed for the Catmull-Clark, modified butterfly and the Loop subdivision schemes. Techniques for assigning material properties to geometric subdivision surfaces are derived, and a motion equation for the dynamic model has been formulated using Lagrangian dynamics. Furthermore, advantages of the physics-based deformable models are incorporated into the conventional subdivision schemes, and a dynamic hierarchical control of this model is introduced. Finally, a multiresolution representation of the control mesh is developed and a unified approach for deriving subdivision surface-based finite elements is presented. The proposed dynamic framework enhances the applicability of the subdivision surfaces in modeling applications. It is also useful for hierarchical shape recovery from large range and volume data sets, as well as for non-rigid motion tracking from a temporal sequence of data sets. Multiresolution representation of the initial mesh controlling the smooth limit surface enables global and local editing of the shape as desired by the modeler. This dynamic framework has also been used for synthesizing natural terrain models from sparse elevation data.
General Note:
Typescript.
General Note:
Vita.
Thesis:
Thesis (Ph.D.)--University of Florida, 1998.
Bibliography:
Includes bibliographical references (leaves 159-167).

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
851387304 ( OCLC )
2427664 ( ALEPH )

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Full Text














A DYNA i',C FRA -.i1l WORK FOR SUBDIVISION SURFACES


CHHANDO.iAY I. iANDAL




















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFIl I'-.i.l:NT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1998




































Copyright 1998
by
C !,!,,i ,,l, I_.,-, ', I,,i ,l I




































To
iy Parents,
and .liy Brother


















ACKNOWLEDGE; 1: NTS


I am deeply grateful to my advisor Dr. Baba C. Vemuri, whose guidance, en-

couragement and support made the completion of this work possible. I thank him

for his academic as well as non-academic advice and assistance during my study at

the University of Florida, and for his impressive example of integrity and academic

responsibility. I wish to express sincere appreciation to my co-advisor, Dr. Hong Qin,

for introducing me to the wonders of computer graphics, for patiently guiding me all

the way till the end, and for assisting me in various ways. I would also like to thank

Dr. Sartaj Sahni, Dr. Paul A. Fi-hl, i, 1 and Dr. Douglas A. Center for serving in

my supervisory committee and advising me on various aspects of this dissertation.

.iy dissertation research was supported in part by the NSF grant ECS-9210648

and the NIH grant ROl-i:'. 'i944 of Dr. Vemuri, and by the NSF CAREER award

CCR-9702103 and DI 1i-9700129 of Dr. Qin. I wish to acknowledge Dr. Tim :.1. il,-

erney, Dr. Gregoire M.il.,iiil.iii Dr. Hughes Hoppe, Dr. Kari Pulli, Dr. Dimitry

Goldgof, Dr. C!, i-i ii., Leonard and Dr. S!,.li.-TII, ig Lai for helping me at various

stages of the work by providing various data sets, software and figures.

I would like to thank the faculty, staff and students of the computer and in-

formation science and engineering department, who were always there to help me.









In my years with the computational vision, graphics and medical imaging group, I

have enjoyed working with a set of bright students, including Yanlin Guo, Yi Cao,

Li C11 ii, Li Wang, 'l illi.iying Huang, Clii.iiir Kapoor, Fengting C11 i Arun Srini-

vasan, Jundong Liu and Jun Ye. Special thanks goes to my long time roommates,

Raja Cli.ii i rjee and Kingshuk .ijii inl.ii. for helping me in various ways during my

stay at Gainesville.

Finilly, I would like to thank my parents and my brother for their constant en-

couragement and support, for cheering me up during the hard days and for providing

an ever increasing incentive to finish my graduate study. I would have been nowhere

near my goals without them.














TABLE OF CONTENTS




ACKNOW LEDG ( il NTS ............................. iv

LIST OF FIGURES ..................... .......... ix

ABSTRACT ............ ..... .......... ...... xiii

CHAPTERS

1 INTRODUCTION ............................... 1

1.1 Problem Statement ............................ 2
1.2 Proposed Solution ................... ......... 3
1.3 Contributions .. .. .. ... .. .. .. .. ... .. .. .. .. .. 5
1.4 Outline of Dissertation .......................... 8

2 BACKGROUND ..................... .......... 11

2.1 Subdivision Surfaces ................... ..... 11
2.2 Physics-based Deformable Surface M'.idels ....... ...... 17
2.3 S~11 ie '.ibdeling Using Physics-based Subdivision-surface '.idel . 20
2.4 11 1iip and '. ition Estimation Using Physics-based Subdivision-surface
'.idel ....... ... .... ....... ...... 22
2.5 Unified Framework for !,liip Recovery and S!liipi dealing g . . 24

3 DYNA' iiC CAT'.IULL-CLARK SUBDIVISION SURFACES ...... ..27

3.1 Overview of the Catmull-Clark Subdivision Scheme . . . ... 27
3.2 Formulation .................. ........... .. .. 30
3.2.1 Assigning Patches to Regular Faces . . . . . ..... 31
3.2.2 Assigning Patches to Irregular Faces . . . . . 33
3.2.3 Kinematics .................. ......... .. 36
3.2.4 Dynamics .................. .......... .. 44
3.2.5 '.i ili!; vel Dynamics ................ .... .. 46
3.3 Finiii Element Implementation ............... . . .. 49
3.3.1 Data Structures .................. ..... .. 49
3.3.2 Normal Elements .................. ..... .. 51
3.3.3 Special Elements .................. ..... .. 51









3.3.4 Force Application .................. ..... .. 52
3.3.5 Discrete Dynamic Equation .................. .. 54
3.3.6 .i, del Subdivision .................. .. .. .. 55
3.4 Generalization of the Approach ............... .. .. 56

4 DYNA-.IIC BUTTER FLY SUBDIVISION SURFACES . . . .... 57

4.1 Overview of the ('.1>dified) Butterfly Subdivision Scheme ...... . 57
4.2 Formulation .................. ........... .. .. 61
4.2.1 Local Parameterization ................. . . .. 62
4.2.2 Dynamics .................. .......... .. 72
4.2.3 `.ililk!vel Dynamics ................ .... .. 74
4.3 Filiii Element Implementation ............... .. ... 76
4.3.1 Elemental '.I1--, and Damping matrices . . . . .... 77
4.3.2 Elemental Stiffness '.,i.i.\ .................. . 79
4.3.3 Other Implementation Issues ..... . . . . . ..... 81
4.4 Generalization of the Approach ............... .. .. 82

5 UNIFIED DYNAI .iC FRAI ii;WORK FOR SUBDIVISION SURFACES 83

5.1 Overview of the Unified Approach . . . . . . . .. .. 83
5.2 Unified Dynamic Framework for Catmull-Clark Subdivision Surfaces 85
5.2.1 Local Parameterization ................. . . .. 88
5.2.2 Dynamics .................. .......... .. 92
5.2.3 Fi;iii Element Implementation . . . . . . .... 92
5.3 Unified Dynamic Framework for Loop Subdivision Surfaces . . 95
5.3.1 Local Parameterization ................. . . .. 97
5.3.2 Dynamics .............. . . . . ... 100
5.3.3 Fi;iii Element Implementation . . . . . . . 100
5.4 A General Outline of the Framework for Approximating Subdivision
Schemes .................. . ............. 101
5.5 A General Outline of the Framework for Interpolatory Subdivision
Schemes ............................. ...... 102

6 'iULTIRESOLUTION DYNAIiCS ............. . . .. .. 103

6.1 Overview of ', ltl i resolution Analysis and Wavelets . . . ... 109
6.2 ':. li ii resolution Analysis for Surfaces of Arbitrary Topology . . 114
6.2.1 .-1 ii Spaces through Subdivision . . . . . . 116
6.2.2 Inner Product ........ . . . . . ........ 118
6.2.3 Biorthogonal Surface Wavelets on Arbitrary .' li l 1 . . .119
6.3 '. li it resolution Representation ...... . . . . . ..125
6.4 Dynamics .................. . . . . ... 128
6.5 Implementation Details .................. .. ...... 129









7 SYSTEM ARCHITECTURE ............


7.1 Topological Information Processing ':. dule ............. 135
7.2 Subdivision '.1, dule ........... ................ 135
7.3 Fiiii Element Analysis '.idule ...... ....... . . . .. 135
7.4 Force Synthesis '.ibdule .................. .. ..... . 136
7.5 Update Engine .................. ......... .. .. 136
7.6 Display '.iodule ............... . . . . . .. . 137

8 APPLICATIONS ................. . . . . ... 138

8.1 Geometric ':. Ideling ......... . . . . . . ... 138
8.2 SliipeI Recovery from Range and Volume Data. . . . . . . 1 1
8.3 Non-rigid M.i )tion Estimation. ............. .. . .. .. 148
8.4 'liili resolution Editing .................. .. ..... . 150
8.5 .'; 1 i 1.! Terrain ':.MIdeling ................... . . 151

9 CONCLUSIONS AND FUTURE WORK .. . . . . . ..... 155

9.1 Conclusions ................. . . . . . ... 155
9.2 Future Directions ............. . . . . . ...... 156
9.2.1 Automatic Cn111, of Topology . . . . . ..... 156
9.2.2 Local Refinement ........ . . . . . . ... 157
9.2.3 Automatic i. del Parameter Selection . . . . .... 157
9.2.4 Constraint Imposition ....... . . . . . ... 157
9.2.5 Recovery of S'1,irp Features ..... . . . . . ..... 158
9.2.6 Automatic o. del Initialization . . . . . ..... 158
9.2.7 Improved Synthesized Force Fields . . . . . . 158

REFERENCES ................... .. ........ ...... 159

BIOGRAPHICAL SKETCH ............ . . . . . ...... 168














LIST OF FIGURES


2.1 Refinement of an initial control mesh to obtain the limit surface.
(Courtesy : H. Hoppe) .................. .. .... .. .. 12

3.1 Catmull-Clark subdivision : (a) initial mesh, (b) mesh obtained after
one step of Catmull-Clark subdivision, and (c) mesh obtained after
another subdivision step. .................. .... 28

3.2 A rectangular mesh and its limit surface consisting of 4 bicubic surface
patches. . . .. .. .. . . . . . . .. .. .. 32

3.3 A mesh with an extraordinary point of degree 3 and its limit surface. 34

3.4 Local subdivision around the extraordinary point and the limit surface. 35

3.5 Local subdivision around the extraordinary point and the correspond-
ing patches in the limit surface from different levels of subdivision. . 43

3.6 T!i, data structure used for dynamic subdivision surface implementation. 50

3.7 '1ldel subdivision to increase the degrees of freedom : (a) evolution
of the initial mesh, and (b) the corresponding limit surface evolution
perceived by the user. ................... ...... . 55

4.1 (a) Ti, control polygon with triangular faces; (b) mesh obtained after
one subdivision step using butterfly subdivision rules. . . . ... 58

4.2 (a) T!i, contributing vertices in the j-th level for the vertex in the j+1-
th level corresponding to the edge between vi and vi; (b) the weighing
factors for different vertices. .................. .... 58

4.3 (a) Tli, weighing factors of contributing vertex positions for an edge
connecting two vertices of degree 6; (b) the corresponding case when
one vertex is of degree n and the other is of degree 6. . . . ... 60

4.4 Ti, smoothing effect of the subdivision process on the triangles of the
initial m esh. . . . . . . . . . .. ... . . . 62









4.5 Tracking a point x through various levels of subdivision : (a) initial
mesh, (b) the selected section (enclosed by dotted lines) of the mesh
in (a), after one subdivision step, (c) the selected section of the mesh
in (b), after another subdivision step. ..... . . . . ...... 64

4.6 Different set of control points at a subdivided level obtained by apply-
ing different subdivision matrices on a given set of control points in a
coarser mesh. .................. ........ . 67- 68

4.7 (a) An initial mesh, and (b) the corresponding limit surface. Tli, do-
mains of the shaded elements in the limit surface are the corresponding
triangular faces in the initial mesh. Tli, encircled vertices in (a) are
the degrees of freedom for the corresponding element. . . . ... 75

5.1 A control mesh with an extraordinary vertex of degree 5 and the corre-
sponding limit surface : (a) using the concepts developed in Cliipi r 3,
where the limit surface consists of quadrilateral normal elements and
a pentagonal special element; (b) using the unified concept developed
in this chapter, where the limit surface consists of one single type of
quadrilateral finite element. .................. .... 86

5.2 In Catmull-Clark subdivision, each non-boundary quadrilateral face in
the control mesh has a corresponding quadrilateral patch in the limit
surface : (a) control mesh, (b) limit surface. . . . . . 87

5.3 (a) Tli, marked 16 control vertices define the shaded quadrilateral
patch associated with the shaded regular face in the control mesh; (b)
the marked 14 control vertices define the shaded quadrilateral patch
associated with the shaded irregular face in the control mesh.. . . 91

5.4 (a) Tli, control polygon with triangular faces; (b) mesh obtained after
one subdivision step using Loop's subdivision rules. . . . ... 96

5.5 An initial mesh and the corresponding limit surface obtained using
Loop's subdivision rules. T!i, domains of the shaded triangular patches
in the limit surface are the corresponding triangular faces in the initial
mesh. Tli, encircled vertices are the control vertices for the corre-
sponding triangular patch in the limit surface. . . . . .... 98

5.6 Each triangular patch in the limit surface can be associated with a
non-boundary triangular face in the initial mesh, which in turn can be
parameterized over a triangle with vertices at (0,0), (1,0) and (0, 1).. 99

6.1 Schematic block diagram of the multilevel dynamics approach. . . 104









6.2 '.i llil, 1vel dynamics vs. multiresolution dynamics . . . . . 106

6.3 Representation of the degrees of freedom in multilevel dynamics and
multiresolution dynamics approach. ..... . . . . . ...... 108

6.4 Tli. filter bank. .................. ........ 112

6.5 Subdivision refinement of a tetrahedron. ..... . . . ..... 119

6.6 Wavelet construction. .................. ........ 123

7.1 System architecture. .................. ........ 134

8.1 (a), (b), (c) and (d) : Initial shapes (obtained applying Catmull-Clark
subdivision rules on control meshes); (e), (f), (g) and (h) : the corre-
sponding modified shapes after interactive force application. .... 139

8.2 (a), (b), (c) and (d) : Initial shapes (obtained applying modified but-
terfly subdivision rules on control meshes); (e), (f), (g) and (h) : the
corresponding modified shapes after interactive sculpting via force ap-
plication.. ................... . . . . ... 141

8.3 (a) Range data of a bulb along with the initialized model, (b) an
intermediate stage of evolution, and (c) the fitted dynamic Catmull-
Clark subdivision surface model. .................. ..143

8.4 (a) Range data of an anvil along with the initialized model, (b) an
intermediate stage of evolution, and (c) the fitted dynamic Catmull-
Clark subdivision surface model. .................. ..144

8.5 (a) Range data of a head along with the initialized model, (d) the
fitted dynamic butterfly subdivision model, and (c) visualization of
the shape from another view point. ..... . . . . . ..... 144

8.6 (a) A slice from a brain '.11:1 (b) initialized model inside the region
of interest superimposed on the slice, (c) the fitted model superim-
posed on the slice, and (d) a 3D view of the dynamic Catmull-Clark
subdivision surface model fitted to the cerebellum. . . . . ... 146

8.7 (a) Data points identifying the boundary of the caudate nucleus on
a I ii: slice of human brain, (b) data points (from all slices) in 3D
along with the initialized model, and (c) the fitted dynamic butterfly
subdivision model. .................. ....... 146









8.8 Snapshots from the animation of canine heart motion over a cardiac
cycle using the dynamic butterfly subdivision model. . . . ... 149

8.9 (a) Ti, initialized model along with the data set; (b) the fitted model
with two subdivisions on the initial mesh, along with attached springs
for editing. Tli~ model after editing (c) at lower resolution, (d) at the
same resolution of the fitted model, and (e) at higher resolution. . 150

8.10 (a) Discrete elevation data set (4096 points), (b) fitted dynamic but-
terfly subdivision surface model with 841 vertices (without noise ad-
dition), and (c) fitted dynamic subdivision surface model with 841
vertices when Gaussian noise is added. ............... ..153

8.11 Synthesized natural terrain of different roughness using the dynamic
butterfly subdivision surface model with 841 vertices from a data set
of 3099 elevation values. ................ ........ 154














Abstract of Dissertation
Presented to the Graduate School of the University of Florida
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy



A DYNAI ilC FRA. 1il:WORK FOR SUBDIVISION SURFACES

By

n1 1 i i ii Id I l l

December 1998


C11.!l ,, ini Dr. Baba C. Vemuri
Cochairman : Dr. Hong Qin
., ij, r Department: Computer and Information Science and Engineering


Subdivision surfaces are extensively used to model smooth shapes of arbitrary

topology. Recursive subdivision on a user-defined initial control mesh generates a

visually pleasing smooth surface in the limit. However, users have to carefully select

the initial mesh and/or manipulate the control vertex positions at different levels

of subdivision hierarchy to satisfy the functional and aesthetic requirements in the

smooth limit surface. This modeling drawback results from the lack of direct manip-

ulation tools for the limit surface. In this dissertation, techniques from physics-based

modeling are integrated with geometric subdivision methodology, and a dynamic

framework is presented for direct manipulation of the smooth limit surface generated

by the subdivision schemes using physics-based "force" tools.









In a typical subdivision scheme, the user starts with an initial control mesh

which is refined recursively using a fixed set of subdivision rules, and a smooth sur-

face is produced in the limit. '.i, I- often this limit surface does not have an analytic

expression, and hence poses challenging problems in incorporating mass and damp-

ing distribution functions, internal deformation energy, forces, and other physical

quantities required to develop a physics-based subdivision surface model. In this

dissertation, local parameterization techniques suitable for embedding the geometric

subdivision surface model in a physics-based modeling framework have been devel-

oped. Specific local parameterization techniques have been fully developed for the

Catmull-Clark, modified butterfly and the Loop subdivision schemes. Techniques

for assigning material properties to geometric subdivision surfaces are derived, and

a motion equation for the dynamic model has been formulated using Lagrangian

dynamics. Furthermore, advantages of the physics-based deformable models are in-

corporated into the conventional subdivision schemes, and a dynamic hierarchical

control of this model is introduced. Fi illy, a multiresolution representation of the

control mesh is developed and a unified approach for deriving subdivision surface-

based finite elements is presented.

Ti, proposed dynamic framework enhances the applicability of the subdivision

surfaces in modeling applications. It is also useful for hierarchical shape recovery

from large range and volume data sets, as well as for non-rigid motion tracking from

a temporal sequence of data sets. ':. ill resolution representation of the initial mesh

controlling the smooth limit surface enables global and local editing of the shape as









desired by the modeler. This dynamic framework has also been used for synthesizing

natural terrain models from sparse elevation data.

















CHAPTER 1
INTRODUCTION



Generating smooth surfaces of arbitrary topology poses a grand challenge for

the computer graphics, computer-aided geometric design and scientific visualization

researchers. Tli, existing techniques for modeling smooth complex shapes can be

broadly classified into two distinct categories namely, (1) explicit patching using

parametric surfaces and (2) subdivision surfaces.

Ti, explicit patching technique involves partitioning the model surface into a

collection of individual parametric surface patches. Adjacent surface patches are

then explicitly stitched together using continuity constraints. This explicit stitching

process is very complicated for modeling smooth surfaces of arbitrary topology due

to the continuity constraints which need to be satisfied along the patch boundaries.

Subdivision surfaces are simple procedural models which offer an alternate rep-

resentation for the smooth surfaces of arbitrary topology. A typical recursive subdi-

vision scheme produces a visually pleasing smooth surface in the limit by repeated

application of a fixed set of refinement rules on an user-defined initial control mesh.

Tli, initial control mesh is a simple polygonal mesh of the same topological type

as that of the smooth surface to be modeled. At each step of subdivision, a finer

polygonal mesh with more vertices and faces is constructed from the previous one via









the refinement process, and the smooth surface is obtained in the limit. However, a

few subdivision steps on the initial mesh generally suffice to approximate the smooth

surface for all practical purposes. Various sets of rules lead to different subdivision

schemes which mainly differ in the smoothness property of the resulting limit surface

and/or the type of initial mesh (i.e. triangular, quadrilateral etc.) chosen.

1.1 Problem Statement

To model a smooth surface of arbitrary topology using subdivision surfaces, first

a polygonal mesh of same topology needs to be chosen as the initial mesh. This ini-

tial mesh, also known as the control mesh, is refined via recursive subdivision using

a fixed set of rules, and a smooth surface of the desired topology is obtained in the

limit. However, users have to carefully select the initial mesh and/or manipulate

the control vertex positions at different levels of subdivision hierarchy in order to

satisfy the functional and aesthetic requirements on the smooth limit surface. For

example, to obtain a desired effect on the smooth limit surface, it might be necessary

to reposition a handful of vertices in the mesh obtained after one subdivision step,

or a large number of vertices might need to be moved in the mesh produced after

three subdivision steps! This process is not intuitive and at best laborious. Despite

the presence of a variety of subdivision schemes in the computer graphics and geo-

metric modeling literature, there is no direct and natural way of manipulating the

limit surface. Ti, current state-of-the-art only permits the modeler to interactively

obtain the desired effects on the smooth surface by kinematically manipulating the

vertex positions at various levels of the subdivision hierarchy. In this dissertation, the









challenging problem of directly manipulating the smooth limit surface at arbitrary

locations/areas is addressed and a novel solution is presented where the modeler can

not only manipulate the smooth limit surface directly but also control the extent of

the manipulation effect (i.e., global or local manipulation) on the limit surface.

Subdivision surfaces have also been used for recovering shapes from a given set of

points in 3D. However, most of the existing subdivision surface-based shape recovery

techniques resort to complex algorithms to derive a mesh for the underlying shape,

and then mesh optimization techniques are used to obtain a compact representation

of the same. t' ., i cheless, this process yields a control mesh of the smooth subdi-

vision surface representing the underlying shape that typically uses a large number

of degrees of freedom (control vertices) for representation. In this dissertation, an

efficient hierarchical method of recovering shapes from range and volume data sets is

proposed where the control mesh of the smooth limit surface will use very few degrees

of freedom for representation.

1.2 Proposed Solution

Physics-based modeling techniques offer a potential solution to the problem

of directly manipulating the smooth limit surface generated by the recursive sub-

division procedure. In the physics-based modeling paradigm, a deformable model

is derived by assigning mass, damping, stiffness and other physical properties to a

surface model. Ti, model is deformed by applying synthesized forces and this de-

formation is governed by physical laws. Now, if the purely geometric subdivision

surface models can be embedded in a physics-based modeling paradigm, then the









smooth limit surface can be deformed directly by using physics-based "force" tools.

However, this procedure-based surface model obtained through the subdivision pro-

cess does not have a closed-form analytic formulation in general, and hence poses

challenging problems to incorporate mass and damping distribution functions, inter-

nal deformation energy, forces, and other physical quantities required to develop a

physics-based model. Techniques of locally parameterizing the smooth limit surface

generated by various subdivision schemes are proposed in this dissertation. Once a

suitable local parameterization scheme is developed for a specific subdivision scheme,

a dynamic framework is provided for the corresponding subdivision scheme where

the modeler can directly manipulate the smooth limit surface by using synthesized

forces. At the same time, a dynamic framework for the wavelets derived using subdi-

vision schemes will assist in adopting a multiresolution representation of the control

mesh defining the smooth limit surface, and the modeler can control the extent of

manipulation effect by choosing a desired level of editing. Synthesized force applica-

tion at a lower resolution will yield a global effect whereas manipulations at a finer

resolution will have localized effects on the limit surface. Tli1 motion of this physics-

based deformable subdivision surface model is governed by a second-order differential

equation, which is solved numerically using the finite element method. N"- types of

finite elements for the chosen subdivision scheme are also presented for representing

the smooth limit surface.

Tli, proposed dynamic framework for the subdivision surfaces provides an effi-

cient solution to the shape recovery problem as well. A simple subdivision surface









model with very few vertices in the control mesh can be initialized (positioned) fully

inside a given set of points in 3D. Ti, initialized model will be deformed by applying

forces synthesized from the given data points. When an equilibrium is obtained, the

number of vertices in the control mesh can be increased via a subdivision step on the

current control mesh thereby increasing the degrees of freedom for model representa-

tion, and a new equilibrium with a better fit to the given data set can be obtained.

This process can be repeated till a prescribed error in fit is achieved. Similar approach

can be taken for shape recovery from volume data sets as well where a different type

of synthesized force needs to be specified. Tli, hierarchical shape recovery process

ensures a compact representation of the recovered smooth limit surface using very

few degrees of freedom.

1.3 Contributions

In this dissertation, techniques from physics-based modeling are integrated with

geometric subdivision methodology to present a scheme for directly manipulating

the smooth limit surface generated by the subdivision process. As a result, unlike

the existing geometric solutions that only allow the operations on control vertices,

the proposed methodology and algorithms permit the user to physically modify the

shape of subdivision surfaces at desired locations via application of forces. This gives

the user a .iiit!" clay/play-dough modeling environment. Ti, proposed model

can be edited directly in a hierarchical fashion using synthesized forces. Also, this

physics-based subdivision surface model efficiently recovers shapes as well as non-

rigid motions from large range and volumetric data sets. Note that this dissertation









neither proposes a new subdivision technique nor provides a different interpretation

of any existing subdivision technique, but integrates the advantages of subdivision

surface-based and physics-based modeling techniques to solve important theoretical

and practical problems. Although the principles of physics-based modeling are well

understood by the graphics experts and modeling researchers, this dissertation will

greatly advance the state of the art in physics-based shape modeling due to the

contributions listed below.


Local parameterization techniques for the smooth limit surface generated by

various subdivision schemes are systematically derived in a hierarchical frame-

work, and subsequently the initial control polyhedron can be viewed as the

parametric domain of the physics-based smooth limit surface.


Tli, smooth dynamic subdivision surface in the limit is treated as a i !"

physical object represented by a set of novel finite elements. Ti, basis (shape)

functions of these new variety of finite elements are derived using the subdivision

schemes. Ti, proposed finite element methods will prove to be useful not only

in the areas of computer graphics and geometric design, but also in engineering

analysis as well.


Ti, subdivision techniques are used to create a surface model that incorporates

mass and damping distribution functions, internal deformation energy, forces,

and other physical quantities. Tli, motion equations are also systematically

derived for this dynamic subdivision surface model whose degrees of freedom are

the collection of initial user-specified control vertices. Tli, i, fi, e, the advantages









of both the physics-based modeling philosophy and the geometric subdivision

schemes are incorporated within a single unified framework.


Users will be able to manipulate this physics-based model in an arbitrary region,

and the model will respond naturally (just like a real-world object would) to

this force application. This shape deformation is quantitatively characterized

by the time-varying displacement of control points that uniquely define the

geometry of the limit surface.


Tlii dynamic framework for wavelets derived using subdivision schemes en-

ables a multiresolution representation of the control mesh defining the smooth

limit surface. This provides additional flexibility to the physics-based modeling

framework since the modeler is free to choose the desired editing level. If a

lower resolution editing level is chosen, the synthesized force application on the

smooth limit surface will have global effects, whereas editing with force-based

tools at a finer resolution will yield a localized effect.


Algorithms and procedures are developed which approximate the proposed new

finite elements using a collection of linear and/or bilinear finite elements subject

to the implicit geometric constraints enforced by the subdivision rules. This

hierarchically-structured approximation can satisfy any user-specified error tol-

erance.


Ti, proposed dynamic framework enhances the applicability of subdivision sur-

face models in various application areas. It provides a direct and intuitive way of









manipulating shapes in a hierarchical fashion for geometric modeling applications.

It has also been successfully used for efficient and hierarchical shape recovery from

range and volume data sets as well as for tracking a shape of interest from a time

sequence of range or volume data sets. Fini lly, the dynamic framework of subdivision

surfaces is combined with a variant of a fractal surface synthesis technique to present

a novel natural terrain modeling method.

1.4 Outline of Dissertation

Tli, rest of the dissertation is organized as follows :

Cli~,ipi r 2 contains an overview of the subdivision surfaces along with a review of

the related literature. Ti, motivation for embedding the subdivision surface models

in a physics-based modeling framework is discussed, and the proposed model is com-

pared with the existing physics-based models. Tli, advantages of a unified framework

for shape modeling and shape recovery are also pointed out in this chapter.

Cllip, ir 3 provides a dynamic framework for the Catmull-Clark subdivision

scheme, which is one of the most popular subdivision techniques for modeling compli-

cated objects of arbitrary genus. Analytic formulation of the limit surface generated

by the Catmull-Clark subdivision scheme is derived and the p!1i, !" quantities

required to develop the dynamic model are introduced [57, 60, 76]. Til, governing

dynamic differential equation is derived using Lagrangian mechanics and is imple-

mented using a finite element technique.









In Cl11,lpi r 4, a dynamic framework is provided for the butterfly subdivision

scheme, another popular subdivision technique for modeling smooth surfaces of arbi-

trary topology. A local parameterization scheme for the butterfly subdivision surface

is derived in a hierarchical style. Tlii physics-based butterfly subdivision model is

formulated as a set of novel finite elements which are optimally approximated by

a collection of standard finite elements subjected to implicit geometric constraints

[58, 61].

Cli,,1pi r 5 presents an unified approach for providing a dynamic framework for

subdivision surfaces in general. In particular, it has been shown that the limit surface

obtained using any subdivision scheme can be viewed as a collection of either quadri-

lateral or triangular finite elements whose basis (shape) functions can be derived

using the chosen subdivision scheme [59].

In Cliipi, r 6, a dynamic framework is presented for the subdivision surface-

based wavelet schemes. This dynamic framework enables a multiresolution represen-

tation of the control mesh defining the smooth limit surface and consequently, the

modeler can control the extent of the effect of manipulation on the limit surface by

choosing a proper editing level.

In Clilpi, r 7, a system that integrates the implementation of the concepts pro-

posed in earlier chapters is presented. Several modules comprise the entire system,

and the functionality of these modules are discussed in this chapter.









Ti, proposed modeling framework is used in various application areas and the

results are presented in Clilipi r 8. Ti, proposed dynamic model has been success-

fully used in geometric modeling, shape recovery and non-rigid motion estimation

applications. A novel technique for natural terrain modeling is also presented where

the dynamic framework is combined with a variant of a fractal surface synthesis

technique.

Finilly, conclusions are drawn in Cliipi, r 9 where future directions of research

are also pointed out.

















CHAPTER 2
BACKGROUND


In this chapter, a brief overview of the subdivision surfaces is presented followed

by a review of the previous work done in the area of subdivision surfaces. This is

followed by an overview of the physics-based models along with a review of the re-

lated literature. Tli, motivating factors for embedding geometric subdivision surface

models in a physics-based modeling paradigm are presented in the next section. TI,

advantages of the proposed dynamic models over the existing physics-based models

for shape recovery are discussed and finally advantages of a unified framework for

shape modeling and shape recovery are also presented.

2.1 Subdivision Surfaces

Tli, input to any subdivision scheme is an initial mesh (also known as control

mesh), \1" = (V, Fo), which is a collection of vertices V0 and a collection of faces

F. Ti, subdivision surface S(.\l") associated with the initial mesh \'1" is defined

as the limit of the recursive application of the refinement R as shown in Fig.2.1.

TIi, refinement R, when applied on a mesh I.k = (V, Fk), creates a refined mesh

\/k+1 = (Vk+l, Fk+l) where the vertices in Vk+1 are computed as affine combinations

of the vertices in Vk and the faces in Fk+1 are obtained by splitting each face in Fk

into a fixed number of sub-faces.
















































(c) Mesh M' = R(R(M)) (d) Limit surface S(M) = R"(M)

Figure 2.1. Refinement of an initial control mesh to obtain the limit surface. (Cour-
tesy : H. Hoppe)


'b) Mesh M-' = R(1M)









It may be noted that the vertices introduced through the subdivision process

have a fixed degree in general (4 in case of quadrilateral meshes and 6 in case of tri-

angular meshes). Ti, vertices which do not have this degree are called extraordinary

vertices. Ti, number of extraordinary vertices are very few as these vertices are not

introduced via subdivision refinement in general. For example, in the Catmull-Clark

subdivision scheme (defined on quadrilateral meshes) the number of extraordinary

vertices does not change after the first subdivision on the initial mesh, whereas in the

modified butterfly subdivision scheme (defined on triangular meshes) the subdivision

process never introduces an extraordinary vertex. Ti, limit surface defined by the

subdivision process is most often C1(first derivative continuous) or C2 (second deriva-

tive continuous) depending on the subdivision rules expect at very few extraordinary

points corresponding to the extraordinary vertices in the mesh. Specific subdivision

schemes along with the properties of their limit surfaces will be discussed in later

chapters.

A lot of research have been carried out on subdivision surfaces, which are mainly

focused either on the development of a new subdivision technique or on analyzing

the smoothness properties of the limit surface generated by a subdivision scheme.

However, in this dissertation the focus is entirely different, namely, embedding the

subdivision surfaces in a physics-based modeling paradigm to enhance the applicabil-

ity of these surface models. In the rest of this section, these i iliid .1 !" techniques

of subdivision surfaces are reviewed.









Cli.il.i i [14] first introduced the concept of subdivision to the computer graphics

community for generating a smooth curve from a given control polygon. During the

last two decades, a wide variety of subdivision schemes for modeling smooth surfaces

of arbitrary topology have been derived following Cli;i .i i's pioneering work on curve

generation. In general, these subdivision schemes can be categorized into two distinct

classes namely, (1) approximating subdivision techniques and (2) interpolatory sub-

division techniques. Tih limit surface obtained using an approximating subdivision

technique only approximates the initial mesh, whereas the limit surface interpolates

the initial mesh in case of interpolatory subdivision techniques.

Among the approximating schemes, the techniques of Doo and Sabin [27, 83]

generalize the idea of obtaining uniform biquadratic patches from a rectangular con-

trol mesh. This scheme is an example of vertex subdivision scheme where a vertex

surrounded by k faces is split into k sub-vertices, one for each face. Doo and Sabin's

subdivision technique can be applied to any mesh of arbitrary topology, and the re-

sulting smooth limit surface would be biquadratic B-splines, except at extraordinary

points corresponding to the extraordinary vertices in the control mesh.

Catmull and Clark [10] developed a method for recursively generating a smooth

surface from a polyhedral mesh of arbitrary topology. Tih Catmull-Clark subdivision

surface, defined by an arbitrary initial mesh, can be reduced to a set of standard

bicubic B-spline patches except at a finite number of degenerate points. This scheme

is an example of face subdivision scheme, where a k-sided face is split into k sub-

faces. Tl, details of this subdivision scheme are discussed in Section 3.1. Halstead









et al. [38] proposed an algorithm to construct a Catmull-Clark subdivision surface

that interpolates the vertices of a mesh of arbitrary topology.

Loop [51] presented a subdivision scheme based on the generalization of quartic

triangular B-splines for triangular meshes. Loop's subdivision rules are presented in

Section 5.3. Hoppe et al. [40] extended Loop's work to produce piecewise smooth

surfaces with discontinuities in selected areas of the limit surface. T11 -- introduced

new subdivision rules that allow for sharp features such as creases and corners in the

limit surface.

Peters and Reif [73] proposed a simple subdivision scheme for smoothing poly-

hedra. Tli, ii refinement rules yield a C1 surface that has a piecewise quadratic pa-

rameterization except at isolated extraordinary points. ':.1- recently, non-uniform

Doo-Sabin and Catmull-Clark surfaces that generalize non-uniform tensor product

B-spline surfaces to arbitrary topologies were introduced by Sederberg et al. [88]. All

the schemes mentioned above generalize recursive subdivision schemes for generating

limit surfaces with a known parameterization. Various issues involved with character

animation using these approximating subdivision schemes were discussed at length

by DeRose et al. [25].

Ti, most well-known interpolation-based subdivision scheme is the ill i, ill '

algorithm proposed by Dyn et al. [30]. Butterfly subdivision method, like other sub-

division schemes, makes use of a small number of neighboring vertices for subdivision.

It requires simple data structures and is extremely easy to implement. However, it

needs a topologically regular setting of the initial (control) mesh in order to obtain









a smooth C1 limit surface. A variant of this scheme with better smoothness prop-

erties can be found in Dyn et al. [29]. Zorin et al. [109] developed an improved

interpolatory subdivision scheme (which we call the modified butterfly scheme) that

retains the simplicity of the butterfly scheme and results in much smoother surfaces

even from irregular initial meshes. T111 interpolatory subdivision schemes have

extensive applications in wavelets on manifolds, multiresolution decomposition of

polyhedral surfaces and multiresolution editing.

A variational approach for interpolatory refinement has been proposed by Kobbelt

[43, 44] and by Kobbelt and Schrider [46]. In this approach, the vertex positions in

the refined mesh at each subdivision step are obtained by solving an optimization

problem. Ti, i, f; e, these schemes are global, i.e., every new vertex position depends

on all the vertex positions of the coarser level mesh. Ti, local refinement property

which makes the subdivision schemes attractive for implementation in the computer

graphics applications is not retained in the variational approach.

Tli, derivation of various mathematical properties of the smooth limit surface

generated by the subdivision algorithms is rather complex. Doo and Sabin [28] first

analyzed the smoothness behavior of the limit surface using the Fourier transform and

an eigen-analysis of the subdivision matrix. Ball and Storry [3, 4] and Reif [80] further

extended Doo and Sabin's prior work on continuity properties of subdivision surfaces

by deriving various necessary and sufficient conditions on smoothness for different

subdivision schemes. Specific subdivision schemes were analyzed by Schweitzer [87],

Habib and Warren [37], Peters and Reif [74] and Zorin [108]. '.1i -, recently, Stam









[90] presented a method for exact evaluation of Catmull-Clark subdivision surfaces

at arbitrary parameter values. It may be noted that the focus of this dissertation

is not on deriving a new subdivision technique or analyzing a subdivision technique,

but on deriving a deformable surface model using these subdivision schemes.

2.2 Physics-based Deformable Surface .odels

SIilipe models can be broadly categorized into two types -lumped parameter

models and distributed parameter models. A lumped parameter model uses small

number of parameters to represent model geometry, whereas a distributed parameter

model uses large number of degrees of freedom for model representation. An example

of a lumped parameter surface model is a superquadric. Spline surfaces serve as a

good example for distributed parameter model.

Deformable surfaces are distributed parameter models which use large number

of degrees of freedom for representing the model geometry. A large variety of shapes

can be modeled using this type of model, but handling a large number of degrees of

freedom can be cumbersome. However, the large degrees of freedom of a deformable

model are embedded in a physics-based framework to allow only a pli -i, .llI mean-

i.hlil model behavior. Various types of energies are assigned to the model using the

degrees of freedom, and the model "d(, I .i ii-" to find an equilibrium state with min-

imal energy. Ti, motion equation is derived using Lagrangian dynamics [36], where

various energies associated with model gives rise to internal and external forces. Ti,

equilibrium state of the model is a model position where the internal deformation

force becomes equal to the externally applied force. Tli, physics-based deformable









models are useful for modeling where the modeler can deform a surface by applying

synthesized forces, and for data fitting where external forces are synthesized from a

given data set such that the model approximates the given data at equilibrium.

A deformable surface model is typically parameterized over the domain [0, 1]2.

Let s(u, v, p) be a deformable surface model, where 0 < u, v < 1 and p is a collection

of the degrees of freedom p i = 1, 2,..., n associated with the model (assuming the

model has n degrees of freedom). Ti, degrees of freedom pi(t) is a set of generalized

coordinates which are functions of time and are assembled into the vector p. Let

fi(t) be the generalized force represented by the vector fp and acting on p Let T be

the kinetic energy, F be the dissipation energy and U be the potential energy of the

deformable surface model. Tir, i, the Lagrangian equation of motion for the model

can be expressed as
d OT OT OF OU
+ --+ f=. (2.1)
dt 0 /; 0 1 (0/ ( 0/,

Let i(u, v) be the mass density function of the surface. Til 11, the kinetic energy

of the surface is



T = f _s'sdudv = pMp, (2.2)
2 i2


where M is called the mass matrix. Similarly, let j(u, v) be the damping density

function of the surface. Tli, I the dissipation energy is



F = 1 'sdudv = pDp, (2.3)
2JJ 2









where D is called the damping matrix. Tli, potential energy of the model can be

defined using the thin-plate-under-tension energy model [96], and is given by


S s' Os Os' Os 02s' 02s
U (a + O22 P11 U
S( u u ov ov o2u o2u
02T 02s 02s' 2s 1
+ 312 v + 322 )dudv = p Kp, (2.4)
uLiv uLiv d2- d2v 2 '


where aii(u, v) and 3ij (u, v) are elasticity functions which control tension and rigidity

respectively of the deformable surface model, and K is known as the stiffness matrix.

Ti, discretized equation of motion can be derived using the expressions of the kinetic,

dissipation and potential energy in Eqn.2.1, which is given by



Mp + Dp + Kp = fp, (2.5)



where fp is the generalized force vector.

Tlii free-form deformable models discussed above were first introduced to com-

puter graphics and visualization in Terzopoulos et al. [99] and further developed by

Terzopoulos and Fleischer [97], Pentland and Williams [72], i i-.11.1,x and Terzopou-

los [65] and Vemuri and Radisavljevic [104]. Celniker and Gossard [11] developed a

system for interactive free-form design based on the finite element optimization of en-

ergy functionals proposed in Terzopoulos and Fleischer [97]. Bloor and Wilson [8, 9],

Celniker and Welch [12] and Welch and Witkin [105] proposed deformable B-spline

curves and surfaces which can be designed by imposing the shape criteria via the

minimization of the energy functionals subject to hard or soft geometric constraints









through Lagrange multipliers or penalty methods. Qin and Terzopoulos [77, 78, 100]

developed dynamic NURBS (D-NURBS) which are very sophisticated models suit-

able for representing a wide variety of free-form as well as standard analytic shapes.

T1i, D-NURBS have the advantage of interactive and direct manipulation of NURBS

curves and surfaces, resulting in physically meaningful hence intuitively predictable

motion and shape variation.

Deformable models are also widely used for shape recovery, segmentation, mo-

tion tracking and other computer vision and medical imaging applications. A detailed

survey of deformable models used in these techniques can be found in ':.1 II1i Iny and

Terzopoulos [64] and the references therein. Some specific existing deformable models

used for shape recovery and non-rigid motion estimation will be reviewed in Section

2.4.


2.3 1,ip,1 .ideling Using Physics-based Subdivision-surface :iodel


Recursive subdivision surfaces are powerful for representing smooth geometric

shapes of arbitrary topology. However, they constitute a purely geometric represen-

tation, and furthermore, conventional geometric modeling with subdivision surfaces

may be difficult for representing highly complicated objects. For example, modelers

are faced with the tedium of indirect shape modification and refinement through time-

consuming operations on a large number of (most often irregular) control vertices

when using typical subdivision surface-based modeling schemes. Despite the advent

of advanced 3D graphics interaction tools, these indirect geometric operations remain

non-intuitive and laborious in general. In addition, it may not be enough to obtain









the most "fair" surface that interpolates a set of (ordered or unorganized) data points.

A certain number of local features such as bulges or inflections may be strongly de-

sired while requiring geometric objects to satisfy global smoothness constraints in

geometric modeling and computer graphics applications. In contrast, physics-based

modeling provides a superior approach to shape modeling that can overcome most of

the limitations associated with traditional geometric modeling approaches. Free-form

deformable models governed by the laws of continuum mechanics are of particular

interest in this context. Ti, -, dynamic models respond to externally applied forces

in a very intuitive manner. Tlii dynamic formulation marries the model geometry

with time, mass, damping and constraints via a force balance equation. Dynamic

models produce smooth, natural motions which are easy to control. In addition, they

facilitate interaction -especially direct manipulation of complex geometries. Fur-

thermore, the equilibrium state of the model is characterized by a minimum of the

deformation energy of the model subject to the imposed constraints. Tli, deformation

energy functionals can be formulated to satisfy local and global modeling criteria, and

geometric constraints relevant to shape design can also be imposed. Tli, dynamic

approach subsumes all of the aforementioned modeling capabilities in a formulation

which grounds everything in real-world physical behavior.

A severe limitation of the existing deformable models, including D-NURBS, is

that they are defined on a rectangular parametric domain. Hence, it can be very

difficult to model surfaces of arbitrary genus using these models. In a recent work,









DeRose et al. [25] assigned material properties to control meshes at various subdivi-

sion levels in order to simulate cloth dynamics using subdivision surfaces. Note that,

they assign physical properties on the control meshes at various levels of subdivision

and not on the limit surface itself, and hence can not solve the modeling goal we

are trying to achieve. In this dissertation, a dynamic framework is presented for

subdivision surfaces which combines the benefits of subdivision surfaces for modeling

arbitrary topology as well as that of dynamic splines for interactive shape manip-

ulation by applying synthesized forces. Ti, proposed dynamic framework presents

the modeler a formal mechanism of direct and intuitive manipulation of the smooth

limit surface, as if they were seamlessly sculpting a piece of real-world !Li A

dynamic framework for subdivision surface-based wavelets adds flexibility in the pro-

posed modeling paradigm. It enables a multiresolution representation of the evolving

control mesh defining the smooth limit surface, and the modeler can control the

extent of manipulation effect by choosing a proper editing level. TI formulation

and implementation details of these dynamic frameworks are discussed in subsequent

chapters.

2.4 11i,11i and otion Estimation Using Physics-based Subdivision-surface : .odel

Tlii dynamic subdivision surface model has been developed primarily for mod-

eling arbitrary (known) topology where modelers can directly manipulate the limit

surface by applying synthesized forces in an intuitive fashion. However, another im-

portant application of the dynamic subdivision surfaces is in non-rigid shape and









motion reconstruction/recovery. Accurate shape recovery requires distributed pa-

rameter models which typically possess a large number of degrees of freedom. On

the other hand, efficient shape representation imposes the requirement of geometry

compression, i.e., models with fewer degrees of freedom. Tli, requirements are

conflicting and numerous researchers have been seeking to strike a balance between

these contradicting requirements [5, 21, 47, 49, 62, 89, 100, 104]. Another important

criterion in the design of shape models is that the initialization of the model during

the shape recovery process should not be restricted to globally parameterized input

meshes since it may be infeasible to globally parameterize shapes of arbitrary topol-

ogy. A physics-based model best satisfying the above mentioned criteria is an ideal

candidate for a solution to the shape recovery problem for obvious reasons.

Deformable models which come in many varieties, have been used to solve this

problem in the physics-based modeling paradigm. T111 models involve the use of

either fixed size [21, 66, 71, 98, 104] or adaptive size [15, 41, 47, 62, 95, 101] grids.

Tli, models with fixed grid size generally use less number of degrees of freedom for

representation at the cost of accuracy of the recovered shape. On the other hand,

models using adaptive grids generally need large number of degrees of freedom to

recover the shapes accurately. Note that the shapes being recovered from the image

data are smooth in most of the medical applications, i.e. the anatomical structures

even with considerable amount of details have more or less a C1 surface. Under

these circumstances, the finite element approaches as in Cohen and Cohen [21] and

.1 i, i i iy and Terzopoulos [62] need a large number of degrees of freedom for deriving









a smooth and accurate representation. In addition, they can not represent shapes

with known arbitrary topology. i'.1 .reover, almost all of these schemes require a

globally parameterized mesh as their input which may be infeasible at times.

Tli, proposed model solves the shape recovery problem very effectively as it

can recover shapes from large range and volume data sets using very few degrees

of freedom (control vertices) for its representation and can cope with any arbitrary

input mesh, not necessarily parameterized. Ti, initialized model deforms under the

influence of synthesized forces to fit the data set by minimizing its energy. Once the

approximate shape is recovered, the model is further subdivided automatically and a

better approximation to the input data set is achieved using more degrees of freedom.

Ti, process of subdivision after achieving an approximate fit is continued till a pre-

scribed error criteria for fitting the data points is achieved. Tli, proposed dynamic

subdivision surface models have also been successfully used in motion tracking and

visualization of a dynamically deforming shape from a time sequence of volumetric

data sets.

2.5 Unified Framework for Sliple Recovery and ,l,1. 1i modeling

Currently, shape recovery and shape modeling are viewed as two distinct areas in

computer graphics and geometric modeling literature. However, there are potential

benefits if these two can be combined in an unified framework. For example, the

modeler starts from scratch to build a specific model in a typical geometric modeling

scenario. First, a rough shape is modeled and then it is fine-tuned by manipulating

control vertex positions to obtain the desired effects. This turns out a cumbersome









process in general. On the other hand, shape recovery using state-of-the-art methods

yield large polygonal meshes which are very difficult to manipulate, especially for

global changes in shape. .- often the resulting meshes from a shape recovery

application are not directly amenable to multiresolution analysis. Computationally

expensive re-meshing techniques are needed to convert these meshes into a specific

type of meshes on which multiresolution analysis can be performed. This specific type

of mesh is known as mesh with lidivision-( .iI II,, 1 i.;l implying a topologically

equivalent mesh with the same connectivity as of the given mesh can be obtained by

recursive subdivision of a very simple known initial mesh.

Tli, proposed dynamic framework combines shape recovery and shape modeling

in an unified framework where the modeler can scan in 3D points of a prototype

model, recover the shape using the proposed dynamic subdivision surface model, and

edit at any desired resolution using physics-based force tools. T!i11- the modeler is re-

lieved of the burden of both building an initial model and editing a cumbersome huge

polygonal mesh. Tli, shape recovery process starts with a smooth subdivision surface

model which has a simple initial mesh. This model is deformed using synthesized

forces from the given data points and is automatically refined using some pre-defined

error in fit criteria. Ti, initial mesh of the final recovered smooth shape has subdivi-

sion connectivity in-built, as it is obtained by subdivision refinement, and therefore

no re-meshing is needed for multiresolution analysis. Tli, dynamic framework for

subdivision surface-based wavelets makes multiresolution editing using physics-based









force tools easy to perform. Ti, -, advantages of shape modeling and shape recovery

in a unified framework will be further illustrated in later chapters.

















CHAPTER 3
DYNA-i, C CAT'.IULL-CLARK SUBDIVISION SURFACES



Subdivision surfaces, as mentioned earlier, can be broadly classified into two

categories -approximating and interpolatory subdivision schemes. Ti, approxi-

mating subdivision schemes typically generalize recursive subdivision techniques for

generating limit surfaces with known parameterizations. In this chapter, a dynamic

framework will be presented for Catmull-Clark subdivision scheme, a popular approx-

imating subdivision technique. T!i, Catmull-Clark subdivision scheme generalizes the

idea of obtaining a bicubic B-spline surface by recursive subdivision of a rectangu-

lar initial mesh. Before discussing the dynamic framework, first the Catmull-Clark

subdivision scheme is briefly reviewed.

3.1 Overview of the Catmull-Clark Subdivision Scheme

Catmull-Clark subdivision scheme, like any other subdivision scheme, starts

with a user-defined mesh of arbitrary topology. It refines the initial mesh by adding

new- vertices, edges and faces with each step of subdivision following a fixed set of

subdivision rules. In the limit, a recursively refined polyhedral mesh will converge to

a smooth surface. Ti, subdivision rules are as follows:























(a) (b)


Figure 3.1. Catmull-Clark subdivision : (a) initial mesh, (b) mesh obtained after one
step of Catmull-Clark subdivision, and (c) mesh obtained after another subdivision
step.



For each face, a new face vertex is introduced which is the average of all the

old vertices defining the face.


For each (non-boundary) edge, a new edge vertex is introduced which is the

average of the following four points: two old vertices defining the edge and two

new face vertices of the faces adjacent to the edge.


For each (non-boundary) vertex V, a new vertex is introduced whose position

is + 2E- (n 3) where F is the average of the new face vertices of all faces
n n n

adjacent to the old vertex V, E is the average of the midpoints of all the edges

incident on the old vertex V and n is the number of the edges incident on the

vertex.


-, "i edges are formed by connecting each newx face vertex to the newx edge

vertices of the edges defining the old face and by connecting each newx vertex









corresponding to an old vertex to the new edge vertices of all old edges incident

on the old vertex.


* i faces are defined as faces enclosed by new edges.


An example of Catmull-Clark subdivision on an initial mesh is shown in Fig.3.1.

Tli, most important property of the Catmull-Clark subdivision surfaces is that a

smooth surface can be generated from any control mesh of arbitrary topology. Tl ,i 1 -

fore, this subdivision scheme is extremely valuable for modeling various complicated

geometric objects of arbitrary topology. Catmull-Clark subdivision surfaces include

standard bicubic B-spline surfaces as their special case (i.e., the limit surface is a

bicubic B-spline surface for a rectangular mesh with all non-boundary vertices of

degree 4). In addition, the aforementioned subdivision rules generalize the recur-

sive bicubic B-spline patch subdivision algorithm. For non-rectangular meshes, the

limit surface converges to a bicubic B-spline surface except at a finite number of ex-

traordinary points. T111 extraordinary points correspond to extraordinary vertices

(vertices whose degree is not equal to 4) in the mesh. Note that, after the first sub-

division, all faces are quadrilaterals, hence all the newx vertices created subsequently

will have four incident edges. Tli, number of extraordinary points on the limit sur-

face is a constant, and is equal to the number of extraordinary vertices in the refined

mesh obtained after applying one step of the Catmull-Clark subdivision on the initial

mesh. Tli, limit surface is curvature-continuous everywhere except at extraordinary

vertices, where only tangent plane continuity is achieved. In spite of the popularity

of Catmull-Clark subdivision surfaces for representing complex geometric shapes of









arbitrary topology, these subdivision surfaces may not be easily parameterizable and

deriving a closed-form analytic formulation of the limit surface can be very difficult.

T111 deficiencies preclude their immediate pointwise manipulation and hence may

restrain the applicability of these schemes. In the rest of the chapter, a dynamic

framework is developed for the Catmull-Clark subdivision surfaces which offers a

closed-form analytic formulation and allows users to manipulate the model directly

and intuitively. It may be noted that recently a scheme was proposed by Stam [90]

for directly evaluating the Catmull-Clark subdivision surfaces at arbitrary parameter

values, but the work presented here on dynamic Catmull-Clark subdivision surfaces

[57, 60, 76] was published prior to the publication of the above-mentioned work.

3.2 Formulation

In this section, a systematic formulation of the new dynamic model based on

Catmull-Clark subdivision surfaces is presented. This dynamic model treats the

smooth limit surface as a function of its initial mesh. However, the control vertex po-

sitions need to be updated continually at every level of subdivision in order to develop

the dynamic framework. Note that, all the vertices introduced through subdivision

are obtained as an affine combination of control vertex positions in the initial mesh.

TI 1, i .,_re, the dynamic behavior of the limit surface can be controlled by formulat-

ing the dynamic model on the initial mesh itself, the only exception being the case

when the initial mesh has non-rectangular faces. This problem can be circumvented

by taking the mesh obtained through one step of subdivision as the initial mesh.

It may be noted that an additional subdivision step may be needed in some cases









to isolate the extraordinary points, and the resulting mesh is treated as the initial

mesh. A typical example of the above mentioned scenario is when the initial mesh is

a tetrahedron where two subdivision steps are needed, and the dynamic framework

can be developed by treating the mesh obtained after two subdivision steps as the

initial mesh. To define the limit surface using the vertices of the initial mesh, the

enumeration of the bicubic patches in the limit surface is necessary. In the next two

sections, various schemes are presented for assigning the bicubic patches to various

faces of the initial mesh.

3.2.1 Assigning Patches to Regular Faces

In Fig.3.2, a rectangular control mesh is shown along with the bicubic B-spline

surface (4 patches) in the limit after an infinite number of subdivision steps. Note

that, each of the bicubic patches in the limit surface is defined by a rectangular face

with each vertex of degree four, thereby accounting for 16 control points (from its

8 connected neighborhood) needed to define a bicubic surface patch in the limit.

Ti, 1, F; re, for each rectangular face in the initial mesh with a degree of 4 at each

vertex, the corresponding bicubic surface patch can be assigned to it in a straight

forward way. In Fig.3.2, the surface patches S1, S2, S3 and S4 are assigned to face

F1, F2, F3 and F4 respectively. Tli, 16 control points for the patch S1, corresponding

to face F1, are highlighted in Fig.3.2. Note that, the initial control mesh can be viewed

as the parametric domain of the limit surface. Tli! i, F, re, face F1 can be thought of

as the portion of the parametric domain over which the patch S1 is defined, i.e., has

non-zero values. t' ,. i. heless, each rectangular face (e.g. Fi) can be parametrically





























































I I -------- ,------- -- ----- r
II I I


-- -" -----


S S
4





--_--- --------------------------------- _
---- ---- -
S2 S3



r----- - - -


Figure 3.2. A rectangular mesh and its limit surface consisting of 4 bicubic surface

patches.









defined over [0, 1]2, and hence, all bicubic B-spline patches defined by 16 control

points are locally parameterized over [0, 1]2.

3.2.2 Assigning Patches to Irregular Faces

In Fig.3.3, a mesh containing an extraordinary point of degree 3 and its limit sur-

face are shown. T11, faces F0, F1, ..., F8 are assigned to bicubic patches So, S1,..., Sg

respectively (as they all have vertices of degree 4) following the aforementioned

scheme. However, the central smooth surface enclosed by the patches So, Si,..., Sg

consists of infinite number of bicubic patches converging to a point in the limit. A re-

cursive way of enumerating these bicubic patches and assigning them to various faces

at different levels need to be developed in order to formulate the dynamic framework

for Catmull-Clark subdivision surface model.

Tli1 idea of enumerating the bicubic patches corresponding to faces having an

extraordinary vertex is shown in Fig.3.4 where a local subdivision of the mesh con-

sisting of faces F0, F1, .., F8, Po, P1, P2 (and not the other boundary faces) of Fig.3.3

is carried out. Topologically, the resulting local subdivision mesh (shown as dotted

mesh) is exactly the same as the mesh in Fig.3.3 and hence exactly the same number

of bicubic patches can be assigned to its faces with vertices of degree 4 as is evident

from Fig.3.4 (the new faces and the corresponding patches are marked by "p" and

"n" respectively). This process of local subdivision and assignment of bicubic patches

around an extraordinary point can be carried out recursively and in the limit, the

enclosed patch corresponding to faces sharing the extraordinary point will converge

to a point. However, there is no need to carry out an infinite number of subdivision































F
II




II
1 I I











S2 S4





30Z) 8 S 6

\ \

S S Si










F - - - - - - - - - -- i o
Fu i









Figure 3.3. A mesh xith an extraordinary point of degree 3 and its limit surface.



















selected mesh for
-- local subdivision


t


Figure 3.4. Local subdivision around the extraordinary point and the limit surface.









steps. This description is for formulation purposes only and the exact implementation

will be detailed in a later section.

3.2.3 Kinematics

In this section, the mathematics for the kinematics of the limit surface is devel-

oped via illustrative examples and then the generalized formulas are presented. It

may be noted that a single bicubic B-spline patch is obtained as the limiting process

of the Catmull-Clark subdivision algorithm applied to an initial 4 by 4 rectangular

control mesh. Let sp(u, v), where (u, v) c [0, 1]2, denote this bicubic B-spline patch

which can be expressed analytically as


3 3
Sp(u, v) (x(u, v), y(u, v), z(u, v))T= E di,jBi,4(u)Bj,4(v), (3.1)
i=0 j=0


where dj represents a 3-dimensional position vector at the (i,j)th control point

location and Bi,4(u) and Bj,4(v) are the cubic B-spline basis functions. Ti, subscript

p on s denotes the patch under consideration. Expressing Eqn.3.1 in a generalized

coordinate system we have

Sp = Jpqp, (3.2)


where Jp is a transformation matrix storing the basis functions of a bicubic B-spline

patch, and is of size (3, 48). Vector q, is the concatenation of all control points

defining a B-spline patch in 3D. Note that in the concatenation of the control points,

each control point has an (x, y, z) component. For example, the (x, y, z) components

of the control point (i,j) correspond to positions 3k, 3k + 1, 3k + 2 where, k =









4i + j respectively in the vector qp. \We can express the entries of Jp explicitly

in the following way: Jp(0, k) = J(1, k + 1) = Jp(2,k + 2) = Bi,4 ()Bj,4(v) and

Jp(0, k + 1) = Jp(0, k + 2) = Jp((1, k) = Jp(, k + 2) = Jp(2, k) = Jp(2, k + 1) = 0.

Limit surface with many bicubic patches from a rectangular initial mesh


Now a limit surface consisting of many bicubic surface patches obtained after

applying an infinite number of subdivision steps to a rectangular initial mesh is

considered. For example, let the limit surface of Fig.3.2 be Sm which can be written

as


1 1 1 1
(u, v) = s, (2u,2v)+s2(2(u- -),2v)+s, (2(u- ),2(v )) +s, (2u, 2(v )),
2 2 2 2

(3.3)

where sm,(2u, 2v) = sm(u,v) for 0 < u, v < and 0 otherwise. Similarly, sm,,Sm3

and s,4 are also equal to sm(u, v) for an appropriate range of values of u, v and 0

outside. It may be noted that sm, sm s,, m ,4 correspond to patches S1, S2, S3, S4

respectively in Fig.3.2. Eqn.3.3 in generalized coordinates becomes


4
Sm= Jlql + J2q2 + J3q3 + J4q4 = iqi, (3.4)
i=1


where Jis are the transformation matrices of size (3, 48) and qis are the (x,y,z) com-

ponent concatenation of a subset of the control points of sm defining sm,, i = 1, 2, 3









and 4. A more general expression for Sm is


4
sm = JiAlq + J2A2qm + J3A3q[m + J4A4qm = JiAiqm = Jmqm. (3.5)
i=1


Where, q, is the 75-component vector of 3D positions of the 25 vertex control mesh

defining the limit surface s,. ':., 111 i'es Ai, 1 < i < 4, are of size (48, 75) each

row consisting of a single nonzero entry (= 1) and the (3, 75)-sized matrix J =

Zi=1 JiAi.

Limit surface with many bicubic patches from an arbitrary initial mesh


Tli, stage is now set to define the limit surface s using the vertices of initial

mesh AM for any arbitrary topology, assuming all faces are quadrilateral and no face

contains more than one extraordinary point as its vertex (i.e., extraordinary points

are isolated). As mentioned earlier, if these assumptions are not satisfied, one or two

steps of global subdivision may be required and the resulting mesh can be treated

as the initial mesh. Let the number of vertices in the initial mesh AM be a, and let

I of these be the extraordinary vertices. Tli, number of faces in the initial mesh are

assumed to be b, and k of these faces have vertices with degree 4 (henceforth termed a

111 I i1i, 1. I ) and each of the remaining (b- k) faces have one of the I extraordinary

vertices (henceforth termed a -"1' i! 1, 1. "). Let p be the 3a = N dimensional vector

containing the control vertex positions in 3D. Using the formulations in Section 3.2.1









and Section 3.2.2, the smooth limit surface can be expressed as


k I
s = ni + sj, (3.6)
i=1 j=l


where ni is a single bicubic patch assigned to each of the normal faces and sj is a

collection of infinite number of bicubic patches corresponding to each of the extraor-

dinary points. As s is a surface defined over the faces of the initial control mesh, each

ni can be viewed as a bicubic patch defined over the corresponding regular face in the

initial control mesh. Similarly, each sj can be defined over the corresponding irregu-

lar faces in the initial mesh (refer to Fig.3.2 -Fig.3.4 for the detailed description on

parametric domains of subdivision surfaces). For the simplicity of the following math-

ematical derivations on the dynamic model, from now on the parametric domain will

not be explicitly provided in the formulation. Employing the same approach taken

before to derive Eqn.3.5, it can be shown that


k k k
n, = (Ji)("pi) = (("Ji)(nAi))p = ("J)p, (3.7)
i=1 i=1 i=1


Where "Ji," pi and "Ai are the equivalent of Ji, pi in Eqn.3.4 and Ai in Eqn.3.5 respec-

tively. Ti, pre-superscript n is used to indicate that these mathematical quantities

describe bicubic patch in the limit surface corresponding to normal faces.

Ti, following notational convention will be used for describing various math-

ematical quantities used in the derivation of the expression for a collection of infinite

number of bicubic patches around an extraordinary vertex. Tli, pre-superscript s









is used to represent a collection of bicubic patches around an extraordinary vertex,

the subscript j is used to indicate the j-th extraordinary point, the post-superscript

represents the exponent of a mathematical quantity and the level indicator (to rep-

resent various levels of subdivision in the local control mesh around an extraordinary

vertex) is depicted via a subscript on the curly braces.

Ti, expression for sj is derived using the recursive nature of local subdivision

around an extraordinary vertex as shown in Section 3.2.2. First, sj can be expressed

as

sj = {SJj}{Spj} + {sj}l, (3.8)


where the first term of Eqn.3.8 is the generalized coordinate representation of the

bicubic B-spline patches corresponding to the normal faces of the new local subdivi-

sion mesh obtained after one subdivision step on the local control mesh (similar to

those patches marked n in Fig.3.4). {sj}1 represents the rest of the infinite bicubic

B-spline patches surrounding the extraordinary point (similar to the central patch

enclosed by patches marked n in Fig.3.4). Tli, vertices in the newly obtained lo-

cal subdivision mesh {'"pj} can be expressed as a linear combination of a subset

of the vertices of the initial mesh M (which will contribute to the local subdivi-

sion) following the subdivision rules. We can name this subset of initial control

vertices {'pj }. Furthermore, there exists a matrix {'Bj}I of size (3c, 3d), such that

{'Bj}{pjJ}0o = {'pj}1 where {"pJ}, and {(PJ}0 are vectors of dimension 3c and 3d

respectively. Applying the idea of recursive local subdivision again on {sj}I, sj can








be further expanded as


sj {'Jj}l{'Bj}l{'pj}o + fJJ2{ Bj}2 + {s}2. (3.9)


In the above derivation, {'Pj}I is a vector of dimension 3d, comprising of a

subset of the vertices defining the 3c dimensional vector {'pj}I. Note that, {'Pj}I

has the same structure as {Spj}o, therefore, there exists a (3d,3d) matrix {'Cj}1

such that {SCj} {spj } = {sPj}l. Each subdivision of a local mesh with d vertices

creates a new local mesh with c vertices which contributes a fixed number of bicubic

B-spline patches. Proceeding one step further, it can be shown that



Sj = {sJj}l{Bj}lp J+pJ}o JJ2{z"BJC}2 Cj SPj}o

+{"J}{SBy C}{s}o + {s,}3. (3.10)


Because of the intrinsic property of the local recursive subdivision around the ex-

traordinary point, we have {"J} = {sJj}2 ... = }{Jj = ... = .Jj. In

addition, the subdivision rules remain the same throughout the refinement process,

we also have {(Bj} = {"Bj}2 = ... = {(Bj}, ... = { Bj},. So, the above

equations can be further simplified leading to



Sj = {sJj} {Bj}l {spj}o {SJjlSBjl {sCj} {spj}o

+{sJj}{SBj "sC} 2 sP }o ...

= sJjl{sBj}l (E {Cj} ){"Spj}o. (3.11)
i=0









Vector Sj can be written as


sj = (SJj)(Spj), (3.12)


where "Jj = {(JjJ}I"BN}I(E o {"CN}i) and "pj = {"pJ}0. Tl, idea of local recursive

subdivision around an extraordinary point is illustrated in Fig.3.5. Note that, each

vertex position in the subdivided mesh is obtained by an affine combination of some

vertices in the previous level and hence any row of {(CjN} sums to 1. Tlii largest

eigenvalue of such a matrix is 1 and it can be shown that the corresponding infinite

series is convergent following a similar approach as in Halstead et al. [38]. Ti, rest

of the derivation leading to an expression for s is relatively straight forward. Using

the same approach used to derive the Eqn.3.7, it can be shown that



Ssj = (sJj)(spj) = ( s(JjA)(sAj))p = (sJ)p. (3.13)
j=1 j=1 j=1


From Eqn.3.6, Eqn.3.7 and Eqn.3.13,



s = (nJ)p + (SJ)p. (3.14)



Let J = ("J) + ("J), and hence


s = Jp.


(3.15)




















fIspi ------




fsB.








1sj

2s so





Figure 3.5. Local subdivision around the extraordinary point and the corresponding
patches in the limit surface from different levels of subdivision.









3.2.4 Dynamics

In the previous section, an expression is derived for the smooth limit surface

obtained via infinite subdivision steps. This expression treats the smooth limit surface

as a function of the control vertex positions in the initial mesh. Now the vertex

positions in the initial mesh defining the smooth limit surface s are treated as a

function of time in order to embed the Catmull-Clark subdivision model in a dynamic

framework. Tlii velocity of this surface model can be expressed as



s(u,v,p) = Jp, (3.16)



where an overstruck dot denotes a time derivative. Ti, physics of the dynamic

subdivision surface model is based on the work-energy version of Lagrangian dynamics

[36] and is formulated in an analogous way to the dynamic framework presented in

Section 2.2.

Let it(u, v) be the mass density function of the surface. T 11, the kinetic energy

of the surface is



T 2= I ,IsTsdudv 2= TMp, (3.17)



where (using Eqn.3.16)

M = if Jdudv (3.18)









is the mass matrix of size (N, N). Similarly, let '(u, v) be the damping density

function of the surface. Ti 1, the dissipation energy is



F I I 'sdudv = Dp, (3.19)



where

D= J JJdudv (3.20)


is the damping matrix of size (N, N).

Ti, potential energy of the smooth limit surface can be expressed as


1
U = p'Kp, (3.21)
2


where K is the stiffness matrix of size (N, N). A thin-plate-under-tension energy

model [96] is used to compute the elastic potential energy of the dynamic subdivision

surface. Ti, corresponding expression for the stiffness matrix K is



K = (/ (a(IJo f + OJJ2 +v 11 + iT 12 J + 322 VJ,)dudv. (3.22)



where the subscripts on J denote the parametric partial derivatives. Ti11 aii(u,v)

and 3ij (u, v)s are elasticity functions controlling local tension and rigidity in the two

parametric coordinate directions. Note that the thin-plate energy expression might

diverge at extraordinary points on the limit surface for Catmull-Clark subdivision

scheme as shown in Halstead et al. [38]. Several methods have been -il-.-. -i ,1 in









Halstead et al. [38] to overcome the problem of the divergent series. However, in this

dissertation, the problem is circumvented by setting the corresponding rigidity coef-

ficients to be zero at these points. Tli11 i fi .e, the thin-plate energy at extraordinary

points is zero. Tli, effect is negligible to the overall thin-plate energy as very few

extraordinary points are present in the smooth limit surface.

Using the expressions for the kinetic, dissipation and potential energy in Eqn.2.1,

the same motion equation as in Eqn.2.5 can be obtained. Ti, generalized force vector

fp, which can be obtained through the principle of virtual work [36], is expressed as



fp = Jff j (, v, t)dudv (3.23)



Different types of forces can be applied on the smooth limit surface, and the limit

surface would evolve over time according to Eqn.2.5 to obtain an equilibrium position

characterized by a minimum of the total model energy.

3.2.5 ',lii i!lvel Dynamics

At this point, a dynamic framework is developed for the Catmull-Clark subdi-

vision scheme where the smooth limit surface evolves over time in response to the

applied forces. Ti, entire process can be described as follows. Given an initial mesh,

a smooth surface is obtained in the limit. Users can directly apply synthesized forces

to this smooth limit surface to enforce various functional and aesthetic constraints.

This direct manipulation is then transferred back as virtual forces acting on the initial

mesh through a transformation matrix (Eqn.3.23), and the initial mesh (as well as









the underlying smooth limit surface) deforms continuously over time until an equilib-

rium position is obtained. Ti, deformation of the surface in response to the applied

forces is governed by the motion equation (Eqn.2.5). Within the physics-based mod-

eling framework, the limit surface evolves as a consequence of the evolution of the

initial mesh. One can apply various types of forces on the limit surface to obtain

a desired effect, but the possible level of details appearing in a shape that can be

obtained through evolution is constrained by the number of control vertices in the

initial mesh. It might be necessary to increase the number of control vertices in the

initial mesh in order to obtain a detailed shape through this evolution process.

Tli, number of control vertices defining the same smooth limit surface can be

increased by simply replacing the initial mesh with a mesh obtained after one subdi-

vision step applied to the initial mesh. This new mesh has more number of vertices

but defines the same limit surface. T!, ,, f, re, the dynamic Catmull-Clark surface

model can be subdivided globally to increase the number of vertices (control points)

of the model. For example, after one step of global subdivision, the initial degrees of

freedom p (refer to Eqn.3.15 and Eqn.3.16) in the dynamic system will be replaced

by a larger number of degrees of freedom q, where q = Ap. A is a global subdivi-

sion matrix of size (.~, N) whose entries are uniquely determined by Catmull-Clark

subdivision rules (see Section 3.1 for the details about the rules). Tlil-, p, expressed

as a function of q, can be written as


p = (ATA) -ATq = Atq,


(3.24)









where At = (A A)- AT. Tli, i, fI .re, Eqn.3.15 and Eqn.3.16 can be rewritten as



s = (JAt)q, (3.25)



and

s(u, v, q) = (JAt)q, (3.26)


respectively. Now the equation of motion needs to be derived for this new subdivided

model involving a larger number of control vertices namely q. Ti, mass, damping

and stiffness matrices need to be recomputed for this "li "I level. T!i, structure of

the motion equation as given by Eqn.2.5 remains unchanged, but the dimensionality

and the entries of M, D, K, p and fp change correspondingly in this newly obtained

subdivided level. In particular the motion equation, explicitly expressed as a function

of q, can be written as

Mq + Dq + Kq = fq, (3.27)


where Mq = ff J/(At)'J' JAtdudv and the derivation of Dq, Kq and fq follow suit.

It may be noted that further subdivision, if necessary, can be carried out in a

similar fashion. T, i i, ,re, multilevel dynamics is achieved through recursive subdi-

vision on the initial set of control vertices. Users can interactively choose the level of

detail representation of the dynamic model as appropriate for their modeling and de-

sign requirements. Alternatively, the system can automatically determine the level of

subdivision most suitable for an application depending on some application-specific

criteria.









3.3 Finii Element Implementation

Ti, evolution of the generalized coordinates for our new dynamic surface model

can be determined by the second-order differential equation as given by Eqn.2.5. An

analytical solution of the governing differential equation can not be derived in general.

However, an efficient numerical implementation can be obtained using the finite ele-

ment method [42]. Tli, limit surface of the dynamic Catmull-Clark subdivision model

is a collection of bicubic patch elements. We use two types of finite elements for this

purpose normal elements (bicubic patches assigned to the normal faces of the initial

mesh) and special elements (collection of infinite number of bicubic patches assigned

to each extraordinary vertex of the initial mesh). Tli, shape functions for the nor-

mal element are the basis functions of the bicubic surface patch, whereas the shape

functions for the special element are the collection of basis functions corresponding

to the bicubic patches in the special element. In the current implementation, the

M, D and K matrices for each individual normal and special elements are calculated

and they can be assembled into the global M, D and K matrices that appear in the

corresponding discrete equation of motion. In practice, we never assemble the global

matrices explicitly in the interest of time performance. Tli, detailed implementation

is explained in the following sections.

3.3.1 Data Structures

Tli, limit surface of the dynamic Catmull-Clark subdivision model is a collection

of bicubic patches, and hence requires us to keep track of the control polygons defining

such patches. We can get the control polygons for the normal elements in the limit










NLL list of faces
List of edges
Subdiv object
-Suv o t list of vertices

list of normal elements


1 c 1 subdivision around
extraordinary vertices




List of faces list of faces
list of edges -u v oj list of edges
-ubdiv object 3ubdiv object
list of vertices list of vertices
List of normal elements -list of normal elements





Figure 3.6. T!i data structure used for dynamic subdivision surface implementation.



surface from the initial control mesh itself. However, we need to locally subdivide the

initial control mesh around the extraordinary vertices to obtain the control polygons

of the bicubic patches in the special elements on the limit surface.

A subdivision surface defined by a control mesh at any level is designed as a

class which has a pointer to its parent mesh, a set of pointers to its offspring meshes

(arising out of local subdivision around the extraordinary vertices at that level), a list

of faces, edges, vertices and normal elements Face, edge, vertex and normal elements

are, in turn, classes which store all the connectivity and other information needed to

either enumerate all the patches or locally subdivide around an extraordinary vertex

in that level. Ti, implementation takes the initial mesh as the base subdivision

surface object (with its parent pointer set to NULL) and locally subdivides the initial

mesh upto a user-defined maximum level around each extraordinary vertex to create









offspring objects at different levels (Fig.3.6). At this point, let's take a closer look at

the normal and special element data structures and computation of the corresponding

local M, D and K matrices.

3.3.2 Normal Elements

Each normal element is a bicubic surface patch and hence is defined by 16

vertices (from the 8-connected neighborhood of the corresponding normal face) in the

initial control mesh. Each normal element keeps a set of pointers to those vertices

of the initial mesh which act as control points for the given element. For a normal

element, the mass, damping and stiffness matrices are of size (16,16) and can be

computed exactly by carrying out the necessary integration analytically. Ti, matrix

J in Eqn.3.18, 3.20 and 3.22 needs to be replaced by Jp (of Eqn.3.2) for computation

of the local M, D and K matrices respectively of the corresponding normal element.

3.3.3 Special Elements

Each special element consists of an infinite number of bicubic patches in the

limit. We have already described a recursive enumeration of the bicubic patches of

a special element in Section 3.2.2. Let us now consider an arbitrary bicubic patch of

the special element in some level j. Ti, mass matrix M, of this patch can be written

as

M, = Mp, (3.28)


where Mp is the normal element mass matrix (scaled by a factor of to take into

account of the area shrinkage in bicubic patches at higher level of subdivision) and









Q2, is the transformation matrix of the control points of that arbitrary patch from the

corresponding control points in the initial mesh. Tli, damping and stiffness matrices

for the given bicubic patch can be derived in a similar fashion. Tli1 -i mass, damping

and stiffness matrices from various levels of (local) subdivision can then be assembled

to form the mass, damping and stiffness matrices of the special element. It may be

noted that the stiffness energy due to plate terms diverges at the extraordinary points

on the limit surface. We solve the problem using a slightly different approach than

the one used in Halstead et al. [38]. When the area of the bicubic patch obtained via

local subdivision of the initial mesh around an extraordinary vertex becomes smaller

than the display resolution, the contribution from such a bicubic patch is ignored in

computing the physical matrices of the corresponding special element. Tli1 number

of extraordinary points in the limit surface is very few, and the above mentioned

approximation is found to work well in practice.

3.3.4 Force Application

Ti!, force f(u, v, t) in Eqn.3.23 represents the net effect of all externally applied

forces. Ti, current implementation supports spring, inflation as well as image-based

forces. However, other types of forces like repulsion forces, gravitational forces etc.

can easily be implemented.

To apply spring forces, a spring of stiffness k can be connected from a point do

to a point xo on the limit surface (or, to the j-th level approximation mesh), the net









applied spring force being


f(x, t) = (do s(x,t))(x- xo)dx, (3.29)



where 8 is the unit impulse function implying f(xo, t) = / (,, s(xo, t)| and vanishes

elsewhere on the surface. However, the 6 function can be replaced with a smooth

kernel to spread the force over a greater portion on the surface. Tli, spring forces can

be applied interactively using the computer mouse or the points from which forces

need to be applied can be read in from a file.

To recover shapes from 3D image data, we synthesize image-based forces. A 3D

edge detection is performed on a volume data set using the 3D '-.1 n;.i-Deriche(. Ii))

operator [69] to produce a 3D potential field P(x, y, z), which we use as an external

potential for the model. Tli, force distribution is then computed as


VP(x" y, z)
f(x, y,z)= A P( (3.30)
|| VPr y, ) ||


where A controls the strength of the force. Tli, applied force on each vertex at the j-

th approximation level is computed by trilinear interpolation for evaluating Eqn.3.23

in Cartesian coordinates. '.i ,re sophisticated image-based forces which incorporate

region-based information such as gradients of a thresholded fuzzy voxel classification

can also be used to yield better and more accurate shape recovery. It may be noted

that we can apply spring forces in addition with the image-based forces by placing









points on the boundary of the region of interest in each slices of the 3D volume (' i :

CT etc.) image data.

3.3.5 Discrete Dynamic Equation

Ti, differential equation given by Eqn.2.5 is integrated through time by dis-

cretizing the time derivative of p over time steps At. Tli, state of the dynamic

subdivision surface at time t + At is integrated using prior states at time t and

t At. An implicit time integration method is used in the current implementation

where discrete derivatives of p are calculated using


(t + At) = pp(t + At) 2p(t) + p(t At) (3.31)
p (t +A t) = 2 (3.31)
At2


and

p(t + At) = p(t + At) p(t At) (3.32)
p(t t)- =t> (3.32)
2At

Ti, elemental mass, damping and stiffness matrices can be assembled to get the

global mass, damping and stiffness matrix for the smooth subdivision surface model.

However, we do not assemble these global sparse matrices explicitly for efficiency

reasons. For the time varying stiffness matrix, we recompute K at each time step.

Using Eqn.2.5, Eqn.3.31 and Eqn.3.32, the discrete equation of motion is obtained as



(2M + DAt + 2At2K)p(t + At) = 2At2f,(t + At) + (DAt 2M)p(t At) + 4Mp(t).

(3.33)











evolLtion evolution



subdiv sion same limit surface
at equili rium with more patches




evol tion evol tion




(a) (b)

Figure 3.7. .idel subdivision to increase the degrees of freedom : (a) evolution of
the initial mesh, and (b) the corresponding limit surface evolution perceived by the
user.


This linear system of equations is solved iteratively between each time step using the

conjugate gradient method [34, 75].


3.3.6 '.iodel Subdivision


Tli, initialized model grows dynamically according to the equation of motion

(Eqn.2.5). Tli, degrees of freedom of the initialized model is equal to the number

of control vertices in the initial mesh as mentioned earlier. When an equilibrium is

achieved for the model, the number of control vertices can be increased by replacing

the original initial mesh by a new initial mesh obtained through one step of Catmull-

Clark subdivision. This increases the number of degrees of freedom to represent the

same (deformed) smooth limit surface and a new equilibrium position for the model

can be obtained. This process is depicted schematically in Fig.3.7. '- idel subdivision

might be needed to obtain a very localized effect on a smooth limit surface. For a









shape recovery application, one may start with a very simple initial model, and when

an approximate shape is recovered, the degrees of freedom can be increased to obtain

a new equilibrium position for the model with a better fit to the given data set. Tl1I

error of fit criteria for the discrete data is based on distance between the data points

and the points on the limit surface where the corresponding springs are attached.

In the context of image-based forces, if the model energy does not change between

successive iterations indicating an equilibrium for the given resolution, the degrees of

freedom for the model can be increased by the above-mentioned replacement scheme

until the model energy is sufficiently small and the change in model energy between

successive iterations becomes less than a pre-specified tolerance.


3.4 Generalization of the Approach

Tli, proposed approach can be generalized for other approximating subdivision

schemes. However, a more general approach is presented in Clplii, r, 5 for deriving the

dynamic framework. Dynamic Catmull-Clark subdivision surface model is reformu-

lated in Section 5.2 using this general approach. A dynamic framework for another

popular approximating subdivision scheme namely, Loop's subdivision scheme, is also

discussed in Cl.,pli r 5.

















CHAPTER 4
DYNAI, iC BUTTERFLY SUBDIVISION SURFACES


In the previous chapter, a dynamic framework has been presented for an approx-

imating subdivision scheme namely, Catmull-Clark subdivision scheme. In this chap-

ter, a dynamic framework is developed for (modified) butterfly subdivision scheme,

which is an interpolatory subdivision technique. First, a brief overview of the (mod-

ified) butterfly subdivision scheme is presented. ':i \, a local parameterization tech-

nique for the limit surface obtained via (modified) butterfly subdivision is discussed.

This parameterization scheme is then used to derive the dynamic model. Tli, imple-

mentation details are also discussed.

4.1 Overview of the ( modifiedd) Butterfly Subdivision Scheme

Tli, butterfly subdivision scheme [30], like many other subdivision schemes used

in geometric design literature/applications, starts with an initial triangular mesh

which is also known as the control mesh. Ti, vertices of the control mesh are known

as the control points. In each step of subdivision, the initial (control) mesh is refined

through the transformation of each triangular face into a patch with four smaller

triangular faces. After one step of refinement, the new mesh in the finer level retains

the vertices of each triangular face in the previous level and hence, interpolates the

coarser mesh in the previous level. In addition, every edge in each triangular face































(a) (b)


Figure 4.1. (a) Ti, control polygon with triangular faces; (b) mesh obtained after
one subdivision step using butterfly subdivision rules.


-W


-W


-W


-W


Figure 4.2. (a) Tli, contributing vertices in the j-th level for the vertex in the j+1-
th level corresponding to the edge between vi and v ; (b) the weighing factors for
different vertices.









is spilt by adding a new vertex whose position is obtained by an affine combination

of the neighboring vertex positions in the coarser level. For instance, the mesh in

Fig.4.1(b) is obtained by subdividing the initial mesh shown in Fig.4.1(a) once. It

may be noted that all the newly introduced vertices corresponding to the edges in the

original mesh have degree 6, whereas the position and degree of the original vertices

do not change in the subdivided mesh.

In the original butterfly scheme, the new vertices corresponding to the edges in

the previous level are obtained using an eight-point stencil as shown in Fig.4.2(a). To

show how this stencil is used, a vertex to be introduced and the contributing vertices

from the stencil are highlighted in Fig.4.1(a). Tli, name of the scheme originated

from the ill i ii Il'-like configuration of the contributing vertices. T11, weighing

factors for different contributing vertex positions are shown in Fig.4.2(b). Ti, vertex

el in the j 1-th level of subdivision, corresponding to the edge connecting vertices

v] and vi at level j, is obtained by



e1 = 0.5(v1 + vj) + 2w(v- + vj) w(vj + v- + v' + vJ), (4.1)



where 0 < w < 1, and vJ denotes the position of the i-th vertex at the j-th level.

Tli, butterfly subdivision scheme produces a smooth C1 surface in the limit

except at the extraordinary points corresponding to the extraordinary vertices (ver-

tices with degree not equal to 6) in the initial mesh [109]. All the vertices introduced

through subdivision have degree 6, and therefore, the number of extraordinary points

in the smooth limit surface equals the number of extraordinary vertices in the initial




































-0.06 0625-w n-2


w
(a) (b)

Figure 4.3. (a) Tli, weighing factors of contributing vertex positions for an edge
connecting two vertices of degree 6; (b) the corresponding case when one vertex is of
degree n and the other is of degree 6.









mesh. Recently, this scheme has been modified by Zorin et al. [109] to obtain better

smoothness properties at the extraordinary points. This modified scheme is known

as modified butterfly subdivision technique. In Zorin et al. [109], all the edges have

been categorized into three classes : (i) edges connecting two vertices of degree 6 (a

10 point stencil, as shown in Fig.4.3(a), is used to obtain the new vertex positions

corresponding to these edges), (ii) edges connecting a vertex of degree 6 and a vertex

of degree n = 6 (the corresponding stencil to obtain new vertex position is shown in

Fig.4.3(b), where q = .75 is the weight associated with the vertex of degree n $ 6,

and si = (0.25 + cos(27ri/n) + 0.5cos(47i/n))/n, i = 0, 1,... 1, are the weights

associated with the vertices of degree 6), and (iii) edges connecting two vertices of

degree n $ 6. Ti, last case can not occur except in the initial mesh as the newly

introduced vertices are of degree 6, and the new vertex position in this last case is

obtained by averaging the positions obtained through the use of stencil shown in

Fig.4.3(b) at each of those two extraordinary vertices.

4.2 Formulation

In this section, a systematic formulation of the dynamic framework for the modi-

fied butterfly subdivision scheme is presented. Unlike the approximating schemes, the

limit surface obtained via modified butterfly subdivision does not have any analytic

expression even in a regular setting. Ti, iI fi 'e, derivation of a local parameterization

suitable for developing the dynamic framework for the limit surface is the key issue

here which is discussed next.












* 0-


Figure 4.4. Tli, smoothing effect of the subdivision process on the triangles of the
initial mesh.


4.2.1 Local Parameterization


Ti, smooth limit surface defined by the modified butterfly subdivision technique

is of arbitrary topology where a global parameterization is impossible. t'. '. i cheless,

the limit surface can be locally parameterized over the domain defined by the initial

mesh following a similar approach described in Lounsbery et al. [53]. Tli, idea

is to track any arbitrary point on the initial mesh across the meshes obtained via

the subdivision process (see Fig.4.4 and Fig.4.5), so that a correspondence can be

established between the point being tracked in the initial mesh and its image on the

limit surface.

Tli, modified butterfly subdivision scheme starts with an initial mesh consisting

of a set of triangular faces. T!i, recursive application of the subdivision rules smoothes

out each triangular face, and in the limit, a smooth surface is obtained which can also

be considered as a collection of smooth triangular patches. Ti, subdivision process

and the triangular decomposition of the limit surface is depicted in Fig.4.4. Note

that, the limit surface can be represented by the same number of smooth triangular

patches as that of the triangular faces in the initial mesh. Tli, i fI .e, the limit surface









s can be expressed as

s = Sk, (4.2)
k=l

where n is the number of triangular faces in the initial mesh and Sk is the smooth

triangular patch in the limit surface corresponding to the k-th triangular face in the

initial mesh.

Tih stage is now set for describing the parameterization of the limit surface over

the initial mesh. Tih process is best explained through the following example. A

simple planar mesh shown in Fig.4.5(a) is chosen as the initial mesh. An arbitrary

point x inside the triangular face abc is tracked over the meshes obtained through

subdivision. Th, vertices in the initial mesh are darkly shaded in Fig.4.5. After

one step of subdivision, the initial mesh is refined by addition of newx vertices which

are lightly shaded. Another subdivision step on this refined mesh leads to a finer

mesh with introduction of newx vertices which are unshaded. It may be noted that

any point inside the smooth triangular patch in the limit surface corresponding to the

face abc in the initial mesh depends only on the vertices in the initial mesh which

are within the 2-neighborhood of the vertices a, b and c due to the local nature of

the subdivision process. For example, the vertex d, introduced after first subdivision

step, can be obtained using the 10 point stencil shown in Fig.4.3(a) on the edge ab.

All the contributing vertices in the initial mesh are within the 1-neighborhood of the

vertices a and b. A 10 point stencil can be used again in the next subdivision step

on the edge db to obtain the vertex g. Some of the contributing vertices at this level

of subdivision, for example, the (lightly shaded) 1-neighbors of the vertex b (except



























(a) (b)


Figure 4.5. Tracking a point x through various levels of subdivision : (a) initial
mesh, (b) the selected section (enclosed by dotted lines) of the mesh in (a), after one
subdivision step, (c) the selected section of the mesh in (b), after another subdivision
step.









d and e) in Fig.4.5(b), depend on some vertices in the initial mesh which are within

the 2-neighborhood of the vertices a, b and c in the initial mesh.

In the rest of the formulation, superscripts are used to indicate the subdivision

level. For example, vuvw, denotes the collection of vertices at level j which control

the smooth patch in the limit surface corresponding to the triangular face uvw at

the j-th level of subdivision. Let vobe be the collection of vertices in the initial mesh

which are within the 2-neighborhood of the vertices a, b and c (marked black in

Fig.4.5(a)). Let the number of such vertices be r. Ti,11 the vector Vab which is the

concatenation of the (x, y, z) positions for all the r vertices, is of dimension 3r. Til, -,

r vertices control the smooth triangular patch in the limit surface corresponding to

the triangular face abc in the initial mesh. Now, there exists four subdivision matrices

(Aabc)t, (Aabc)l, (Aabc)r and (Aabc)m of dimension (3r, 3r) such that



Vadf = (Abc\Vc

Vbed = (Aabc)lVabc

f = (eAabc)rVbc

Vde = (Aabc),mVb0, (4.3)



where the subscripts t, 1, r and m denote top, left, right and middle triangle positions

respectively (indicating the relative position of the new triangle with respect to the

original triangle), and V Vd, ved, Vfe and vef are the concatenation of the (x, y, z)

positions for the vertices in the 2-neighborhood of the corresponding triangle in the









newly obtained subdivided mesh. Ti. new vertices in this level of subdivision are

lightly shaded in Fig.4.5(b). Ti, 2-neighborhood configuration of the vertices in the

newly obtained triangles is exactly the same as that of the original triangle, hence

local subdivision matrices are square and the vector dimensions on both sides of

Eqn.4.3 are the same. This concept is further illustrated in Fig.4.6.

Carrying out one more level of subdivision, a new set of vertices which are

unshaded in Fig.4.5(c) are obtained along with the old vertices. Adopting a similar

approach as in the derivation of Eqn.4.3, it can be shown that



V gi = (Abed)tV d

v 2g = (Abed)V Ied

Veih = (Abed)Ved

Vghi = (Abed)mVbed (4.4)



Tli, relative position of the triangular face dgi in Fig.4.5(c) with respect to the

triangular face bed is topologically the same as of the triangular face adf in Fig.4.5(b)

with respect to the triangular face abc. Ti, I r ;.e, one can write (Abed)t = (Aabc)t.

Using similar reasoning, Eqn.4.4 can be rewritten as



V gi = (Abed)tV d = (Aabc)tV d

V hg = (Abed)V Ied = (Aabc)V Ied

Vi = (Abed)rVed = (Aabc)rV1ed
Veih __bd






67






IV,-bc








abc l I 1









eb




------------------------------



(a)





(Aabc)










1 /








(b)























































Figure 4.6. Different set of control points at
different subdivision matrices on a given set


a subdivided level obtained by applying
of control points in a coarser mesh.









vi = (Abed),Vd = (Aabc),Ved. (4.5)



Combining Eqn.4.3 and Eqn.4.5, it can be shown that



dgi = (Aabc)t(Aabc)tVbc,

Vhg = (Aabc)t(Aabc)tVbc,

2 0
Veih = (Aabc)r(Aabc)lVabc,

Vhi = (Aabc)m(Aabc)tVbc. (4.6)



Let x be a point with barycentric coordinates (Oabc( 0bc 7abc) inside the trian-

gular face abc. When the initial mesh is subdivided, x becomes a point inside the

triangular face bed with barycentric coordinates ( bd, led, Tbed). Another level of

subdivision causes x to be included in the triangular face dgi with barycentric coor-

dinates (Ogi,, q2g 7ygi). Let sbc denote the j-th level approximation of the smooth

triangular patch Sabc in the limit surface corresponding to the triangular face abc in

the initial mesh. Now vbc can be written as


r r r
oabc [=ax, bx, C, ..., ay, by, cy, . ., az, bz, c, ....



where the subscripts x, y and z indicate the x, y and z coordinates respectively of the

corresponding vertex position. Tli, expressions for vd and v 2 can also be written
~h~1~311V1 1 II bed dg









in a similar manner. 'i \ the matrix BOb can be constructed as follows:

r r r
bc ybc, 0,..., 0, 0,... 0,..., 0
r r r
B xabc, ( 0,.,00 1 abc,, c bc, O, O, 0...,0
0 0, "", "'
r r r
0,...,0,0,...,0, ab bc bc, ...
(2.)ab, V ^c. .... abc, b


Tli, matrices Bed and B' can also be constructed in a similar fashion. Now so~ (x),

sbc((x), and sb c(x) can be written as


sabc(x) 0 Bbc(X)Vbc,

bc(x) = Bbed(x)vbed = B (x)(Aabc)V bc,

sbc(x) = B ( (x)v = B 0 j(x)(Aabc,)tved = B j (x)(Abc)(Aabc)V b
(4.7)




Proceeding in a similar way, the expression for sbc(x), j-th level approximation

of Sabc (x), is given by



sbc(x) = BL,,(x) (Aabc)m ... (Aabc)t(Aabc); Vabc

= BI(x) (Abc)Vabc

S bc (X) V abc, (4.8)



where x is inside the triangular face uvw at level j (with an assumption that uvw is the

triangular face in the middle with respect to its coarser level original triangular face in









the previous level), (A.bc) = (A.abc (Aabc),(Aabc) and Bbc (x) = BW,(x)(AbJ).

It may be noted that the sequence of applying (Aabc)t, (Aabc)l, (Aabc), and (Aabc),

depends on the triangle inside which the tracked point x falls after each subdivision

step. Fin.illy, the local parameterization process can be completed by writing



Sabc(x) = (lim Bjbc(X))V bc = Babc(X)V.be. (4.9)



In the above equation, Babc is the collection of basis functions at the vertices of

vabc. It may also be noted that the modified butterfly subdivision scheme is a sta-

tionary subdivision process, and hence new vertex positions are obtained by affine

combinations of nearby vertices. This guarantees that each row of the matrices

(Aabc)t, (Aabc)l, (Aabc)r and (Aab(c), sums to one. Tli, largest eigenvalue of such ma-

trices is 1 and therefore the limit in Eqn.4.9 exists. Now, assuming the triangular

face abc is the k-th face in the initial mesh, Eqn.4.9 can be rewritten as



Sk(x) = Bk(x)v = Bk(x)Akp, (4.10)



where p is the concatenation of the (x,y,z) positions of all the vertices in the initial

mesh and the matrix Ak, when post-multiplied by p, selects the vertices vk controlling

the k-th smooth triangular patch in the limit surface. If there are t vertices in the

initial mesh and r of them control the k-th patch, then p is a vector of dimension 3t,

Ak is a matrix of dimension (3r, 3t), and Bk(x) is a matrix of dimension (3, 3r).









Combining Eqn.4.2 and Eqn.4.10, it can be shown that



s(x) = (E Bk(x)Ak)p = J(x)p, (4.11)
k=l


where J, a matrix of dimension (3, 3t), is the collection of basis functions for the

corresponding vertices in the initial mesh. Ti, vector p is also known as the degrees

of freedom vector of the smooth limit surface s.

4.2.2 Dynamics

An expression of the limit surface obtained via modified butterfly subdivision

process is derived in the last section. Now a dynamic framework for the limit surface

can be derived using this expression. Tli, derivation of the dynamic model from this

point is exactly similar in spirit as that of the dynamic Catmull-Clark subdivision

model presented in Section 3.2.4. Ti, vertex positions in the initial mesh defining

the smooth limit surface s are treated as a function of time in order to embed the

modified butterfly subdivision model in a dynamic framework. Ti, velocity of this

surface model can be expressed as



s(x, p) = J(x)p, (4.12)



where an overstruck dot denotes a time derivative and x E So, So being the domain

defined by the initial mesh. As in the case of the dynamic Catmull-Clark subdivision

surfaces, the Lagrangian equation of motion (Eqn.2.1) is used to derive the dynamic

modified butterfly subdivision surface model.









Let p be the mass density function of the surface. Ti 1, the kinetic energy of

the surface is


r 1.
T (x)s(x)s(x)x Mp, (4.13)
2 xESO 2


where (using Eqn.4.12) M = fXo t(x)JT(x)J(x)dx is the mass matrix of dimen-

sion (3t, 3t). Similarly, let 7 be the damping density function of the surface. T!i,

dissipation energy is


1r 1.
F I (x)s (x)s(x)dx = Dp, (4.14)
2 Jxso 2


where D = fXEso y(x)JT(x)J(x)dx is the damping matrix of dimension (3t, 3t). Tli,

potential energy of the smooth limit surface can be expressed as


1
U = p'Kp, (4.15)
2


where K is the stiffness matrix of dimension (3t, 3t) obtained by assigning various

internal energies to a discretized approximation of the limit surface and is detailed

in Section 4.3.

Using the expressions for the kinetic, dissipation and potential energy in Eqn.2.1,

the same motion equation as in Eqn.2.5 can be obtained. Tli, generalized force vector









fp can be obtained is a similar fashion described in Section 3.2.4 and is given by



fp= I J (x)f(x, t)dx. (4.16)



4.2.3 Ai li ilvel Dynamics

Ti, initial mesh of the dynamic modified butterfly subdivision surface model

can be subdivided to increase the degrees of freedom for model representation. Ti11

situation is exactly similar to that of the dynamic Catmull-Clark subdivision surface

model as described in Section 3.2.5 and a similar example is used for illustration.

After one step of modified butterfly subdivision, the initial degrees of freedom p (refer

to Eqn.4.11 and Eqn.4.12) in the dynamic system is replaced by a larger number of

degrees of freedom q, where q = Bp. B is a global subdivision matrix of size (3s, 3t)

whose entries are uniquely determined by the weights used in the modified butterfly

subdivision scheme (see Section 4.1 for the weights). Tlit-, p, expressed as a function

of q, can be written as

p = (BTB)-IBTq = B'q, (4.17)


where Bt = (B 'B)-1B'. Tli, i, ._re, Eqn.4.11 and Eqn.4.12 is modified as



s(x) = (J(x)Bt)q, (4.18)



and


(4.19)


s(x, q) = (J(x)Bt)q,



































Figure 4.7. (a) An initial mesh, and (b) the corresponding limit surface. Ti, domains
of the shaded elements in the limit surface are the corresponding triangular faces in
the initial mesh. Ti, encircled vertices in (a) are the degrees of freedom for the
corresponding element.


respectively. Tli, motion equation, explicitly expressed as a function of q, can be

written as


M~iq + Dq + Kqq = fL,


where Mq


(4.20)


fxst 1(x)(Bt)TJT(x)J(x)Btdx, S1 being the domain defined by the


newly obtained subdivided mesh. Ti, derivations of D,, K, and f, are similar.









4.3 Fini, Element Implementation

In the previous section, the smooth limit surface obtained via modified butterfly

subdivision process is expressed as a function of the control vertex positions in its

initial mesh; mass and damping distribution, internal deformation energy and forces

are assigned to the limit surface in order to develop the corresponding physical model.

In this section, the implementation of this physical model is detailed. It may be noted

that even though the formulation of the dynamic framework for modified butterfly

subdivision scheme is similar to that of Catmull-Clark subdivision scheme once the

local parameterization is derived, the finite element implementation is totally different

as the limit surface does not have any analytic expression in case of the modified

butterfly subdivision scheme.

In Section 4.2 it was pointed out that the smooth limit surface obtained by the

recursive application of the modified butterfly subdivision rules can be represented by

a set of smooth triangular patches, each of which is represented by a finite element.

Tli, shape (basis) function of this finite element is obtained by smoothing a hat

function through repeated application of the modified butterfly subdivision rules.

Tli, number of finite elements in the smooth limit surface is equal to the number

of triangular faces in the initial mesh as mentioned earlier (refer Fig.4.4 and 4.7).

A detailed discussion on how to derive the mass, damping and stiffness matrices

for these elements is presented next. Tli, elemental matrices can be assembled

to obtain the global physical matrices M, D and K, and a numerical solution to

the governing second-order differential equation as given by Eqn.2.5 can be obtained









using finite element analysis techniques [42]. Tli, same example as in Section 4.2

(refer Fig.4.5) is used to develop the related concepts. Tli, concept of elements along

with the control vertices and their corresponding domains is further illustrated in

Fig.4.7.

Now it will be shown how to derive the mass, damping and stiffness matrices

for the element corresponding to the triangular face abc in Fig.4.5. Tli, derivations

hold for any element in general.

4.3.1 Elemental .I,--, and Damping matrices

Ti, mass matrix for the element given by Sab, (Eqn.4.9) can be written as



Mabc = i (x) s Babc(x) (B(x)T Bab (x) }dx. (4.21)



However, from Eqn.4.9 it is known that the basis functions corresponding to the

vertices in vabe which are stored as entries in Babc are obtained as a limiting process.

Ti, -, basis functions do not have any analytic form, hence computing the inner

product of such basis functions as needed in Eqn.4.21 is a challenging problem. In

Lounsbery et al. [53], an outline is provided on the computation of these inner

products without performing any integration. In this dissertation, a different yet

simpler approach is developed to solve this problem. Tli, smooth triangular patch in

the limit surface corresponding to the face abc in the initial mesh is approximated by

a triangular mesh with 4J faces obtained after j levels of subdivision of the original

triangular face abc (each subdivision step splits one triangular face into 4 triangular









faces). Ti, ii the mass matrix can be expressed as


4J
Mabc = : ( t(x){BNbc(x) } { Bbc(X) }dx. (4.22)
i=1 xEAi


Ti, j-th level approximation of the corresponding basis functions can be explicitly

evaluated (refer Eqn.4.8 for an expression of B bc). An important point to note is

that Eqn.4.8 involves several matrix multiplications and hence can be very expensive

to evaluate. However, the matrix (Ab,)(= (Aabc)m .. (Aabc),(Aabc)1) in Eqn.4.8

encodes how vertices in the 2-neighborhood of the triangular face uvw at level j are

related to the vertices in the 2-neighborhood of the triangular face abc in the initial

mesh. In the implementation, how a new vertex is obtained from the contributing

vertices in its immediate predecessor level is tracked. TI,, information stored in (Aabc)

can be obtained by tracking the contributions from level j to level 0 recursively. TI [ -.

the entries of (Abc) can be determined without forming any local subdivision matrices

and thereby avoiding subsequent matrix multiplications.

By choosing a sufficiently high value of j, a reasonably good approximation of

the elemental mass matrices is achieved. Tli, computations involved in evaluating

the integrals in Eqn.4.22 are eliminated by using discrete mass density function which

has non-zero values only at the vertex positions of the j-th subdivision level mesh.

TI! 1, fi .e, the approximation of the mass matrix for the element can be written as


k
Mabc= E (v)L{Bbc(v )} b { c(vi)}, (4.23)
i=1









where k is the number of vertices in the triangular mesh with 4- faces. This approxi-

mation has been found to be very effective and efficient for implementation purposes.

Tli, elemental damping matrix can be obtained in a similar fashion.

4.3.2 Elemental Stiffness '.i i7,\

Ti, internal energy is assigned to each element in the limit surface, thereby

defining the internal energy of the smooth subdivision surface model. A similar

approach as in the derivation of the elemental mass and damping matrix is taken

and the internal energy is assigned to a j-th level approximation of the element.

Three types of internal energy are used -tension, stiffness and spring energy.

For the examples used throughout the paper, this energy at the j-th level of approx-

imation can be written as



Eab, Eb, = (E b,)t + (Eabt + (Eabc)sp, (4.24)



where the subscripts t, st and sp denote tension, stiffness and spring respectively.

Tli, expression for the tension energy, which is essentially equivalent to the first

order strain (membrane) energy [96], is



1 2
(abc) 2 v -vI

= 2I b} (Kbc)IVbc}, (4.25)



where kt is a constant, v, and v the 1-th and m-th vertex in the j-th level mesh,

are in the 1-neighborhood of each other, Q is the domain defined by all such vertex









pairs, and vb, is the concatenation of the (x,y,z) positions of all the vertices in the

j-th subdivision level of the triangular face abc in the initial mesh.

Similarly, the expression for stiffness energy, which is equivalent to the second

order strain (thin plate) energy [96], can be written as



1 j
Eab)St 2kst IV] 2v + v |
2 n
= 1vk,,u} (Ku )s{v,}, (4.26)



where v{, vi and vi are the three vertices of a triangular face. Ti, summation in-

volves three terms corresponding to each triangular face, and is over all the triangular

faces in the mesh at the j-th level of subdivision.

Fiiilly, a spring energy term is added which is given by


1 (km),,p(I|v v~| 1m,)
(EIp = (vE v )- v
(abcp 2 vm
= 1{v,} (K ){vb}, (4.27)



where (kim), is the spring constant, Q is the domain as in Eqn.4.25 and tm is the

natural length of the spring connected between v1 and v-. It may be noted that

the entries in (K b,) depend on the distance between the connected vertices and

hence, (K b,) unlike other elemental matrices, is a function of time which needs to

be recomputed in each time step.









Combining the expressions for tension, stiffness and spring energy, it can be

shown that



EJb =c {vb( {(K b)t + k. I (K~).(t+ (K bc) sp Vbc}
= {vbc } (Kabca){v bta
1 2 I

= {(Ab)vb.}} (Kb)(Ab bc}J

= {vb}T {(Ajb)Tc(K 'b)(Abc)}{ }, (fI



where (Aabc) and vabe are same as in Eqn.4.8. T, i, f; re, the expression for the

elemental stiffness matrix is given by



Kabc = (AbT (Kjb)(Ajb). (4.29)



It may be noted that the matrix multiplications for constructing Kabc are avoided in

the implementation by following the same technique described in Section 4.3.1.

4.3.3 Other Implementation Issues

Tli, techniques of applying synthesized forces on the smooth limit surface ob-

tained via modified butterfly subdivision process are similar to those described in

Section 3.3.4 in the context of dynamic Catmull-Clark subdivision surface model.

Ti, techniques of model subdivision and discrete dynamic equation derivation are

also the same.









4.4 Generalization of the Approach

Ti, proposed approach of deriving the dynamic framework for modified butter-

fly subdivision scheme is generalized for other interpolatory subdivision schemes in

the next chapter, Cli.,11i r 5.


















CHAPTER 5
UNIFIED DYNA I C FRA 1ii;WORK FOR SUBDIVISION SURFACES



Dynamic framework for specific subdivision schemes was presented in the last

two chapters. In Clilipi r 3, Catmull-Clark subdivision scheme, a representative of

the approximating subdivision schemes, is embedded in a physics-based modeling en-

vironment. In C(ri _iir 4, a dynamic framework is provided for the modified butterfly

subdivision scheme, a popular interpolatory subdivision technique. In this chapter,

a unified approach is presented for embedding any subdivision scheme in a dynamic

framework.

5.1 Overview of the Unified Approach

Tli, key concept of the unified approach is the ability to express the smooth

limit surface generated by a subdivision scheme as a collection of a single type of

finite elements. Ti, type of this finite element depends on the subdivision scheme

involved. For example, the limit surface generated by the Catmull-Clark subdivision

scheme can be expressed as a collection a quadrilateral finite element patches. This

approach differs form the one taken in Cliip i, 3 where the limit surface was a

collection of normal elements (corresponding to the quadrilateral bicubic B-spline

patches away from the extraordinary points) and special elements (corresponding to









the n-sided patches near extraordinary vertices of degree n). This concept will be

further elaborated in the next few sections.

Tli, unified approach adopts two different strategies depending on the subdivi-

sion scheme involved. One strategy is for the approximating schemes, whose limit

surface is some type of B-spline surface away from the extraordinary points, and

the other is for the interpolatory subdivision schemes, where the limit surface does

not have any analytical expression. It may be recalled that recursive subdivision is

a method for smoothing polyhedra, and hence corresponding to each n-sided face

in the control mesh, there would be a smooth n-sided patch in the limit surface

in general. For example, the control mesh (after at most one subdivision step) of

the Catmull-Clark subdivision scheme consists of quadrilateral faces, which when

smoothed through subdivision lead to quadrilateral patches in the limit surface (see

Fig.5.1(b)). In the case of approximating subdivision schemes, the faces which do

not contain any extraordinary vertex lead to spline patches with analytic expressions.

On the other hand, faces which have one extraordinary vertex, lead to patches whose

analytic expressions are difficult to determine. Continuing the example with the

Catmull-Clark subdivision scheme, the faces without extraordinary vertices lead to

quadrilateral bicubic B-spline patches in the limit, whereas the faces with one extraor-

dinary vertex lead to quadrilateral patches whose analytic expressions are difficult

to obtain. In Cliipl r 3, n such adjacent patches corresponding to an extraordinary

vertex of degree n are effectively combined together to obtain an expression for the re-

sulting n-sided patch (see Fig.5.1(a)). Recently, Stam [90, 91] has introduced schemes









for exactly evaluating the limit surface, even near the extraordinary points, for ap-

proximating subdivision schemes. Analytic expressions for the patches resulting from

smoothing faces with an extraordinary vertex can be obtained using these schemes.

T!i, 1, fi .e, the limit surface can be expressed as a collection of a single patch type,

and consequently a single type of finite element can be used to provide the dynamic

framework for the approximating subdivision schemes. This unified framework for

approximating subdivision schemes is fully worked out in this chapter for Catmull-

Clark and Loop subdivision schemes and a general outline is also presented on how

to carry out the same strategy for other approximating subdivision schemes.

Tli1 limit surface resulting from an interpolatory subdivision scheme does not

have an analytic expression in general, and hence a strategy similar to the one adopted

in Cliipi, r 4 needs to be used. An outline of the strategy to be used for interpolatory

subdivision schemes is also presented in this chapter.

5.2 Unified Dynamic Framework for Catmull-Clark Subdivision Surfaces

A systematic formulation along with the implementation details of the unified

dynamic framework for Catmull-Clark subdivision surfaces is presented in this sec-

tion. Tli1 key difference between the framework developed in Clilipi r 3 and the one

presented here is the representation of the limit surface, which leads to different types

of finite elements used for developing the dynamic framework. This is illustrated with

a schematic diagram in Fig.5.1.




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A DYNAMIC FRAMEWORK FOR SUBDIVISION SURFACES
By
CHHANDOMAY MANDAL
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1998

©Copyright 1998
by
Chhandomay Mandal

To
My Parents,
and My Brother

ACKNOWLEDGMENTS
I am deeply grateful to my advisor Dr. Baba C. Vomuri, whose guidance, en¬
couragement and support made the completion of this work possible. I thank him
for his academic as well as non-academic advice and assistance during my study at
the University of Florida, and for his impressive example of integrity and academic
responsibility. I wish to express sincere appreciation to my co-advisor, Dr. Hong Qin,
for introducing me to the wonders of computer graphic's, for patiently guiding me all
the way till the end, and for assisting me in various ways. I would also like to thank
Dr. Sartaj Sahni, Dr. Paul A. Fishwick and Dr. Douglas A. Conzor for serving in
my supervisory committee and advising me on various aspects of this dissertation.
My dissertation research was supported in part by the NSF grant ECS-9210648
and the NTH grant RO1-LM05944 of Dr. Vomuri, and by the NSF CAREER award
CCR-9702103 and DMI-9700129 of Dr. Qin. I wish to acknowledge Dr. Tim Mcln-
ornoy, Dr. Grogoiro Malandain, Dr. Hughes Hoppe, Dr. Kari Pulli, Dr. Dimitry
Goldgof, Dr. Christina Leonard and Dr. Shang-Hong Lai for helping mo at various
stages of the work by providing various data sots, software and figures.
I would like to thank the faculty, staff and students of the computer and in¬
formation science and engineering department, who wore always there to help mo.
IV

In my years with the computational vision, graphic's and medical imaging group, I
have enjoyed working with a set of bright students, including Yanlin Guo, Yi Gao,
Li Chen, Li Wang, Shuangying Huang, Chinar Kapoor, Fengting Chen, Arun Srini-
vasan, Jundong Liu and Jun Ye. Special thanks goes to my long time roommates,
Raja Chatterjee and Kingshuk Majumdar, for helping me in various ways during my
stay at Gainesville.
Finally, I would like to thank my parents and my brother for their constant en¬
couragement and support, for cheering me up during the hard days and for providing
an ever increasing incentive to finish my graduate study. I would have been nowhere
near my goals without them.

TABLE OF CONTEXTS
ACKNOWLEDGMENTS iv
LIST OF FIGURES ix
ABSTRACT xiii
CHAPTERS
1 INTRODUCTION 1
1.1 Problem Statement 2
1.2 Proposed Solution 3
1.3 Contributions 5
1.4 Outline of Dissertation 8
2 BACKGROUND 11
2.1 Subdivision Surfaces 11
2.2 Physics-based Deformable Surface Models 17
2.3 Shape Modeling Using Physics-based Subdivision-surface Model ... 20
2.4 Shape and Motion Estimation Using Physics-based Subdivision-surface
Model 22
2.5 Unified Framework for Shape Recovery and Shape Modeling 24
3 DYNAMIC CATMULL-CLARK SUBDIVISION SURFACES 27
3.1 Overview of the Catmull-Clark Subdivision Scheme 27
3.2 Formulation 30
3.2.1 Assigning Patches to Regular Faces 31
3.2.2 Assigning Patches to Irregular Faces 33
3.2.3 Kinematics 36
3.2.4 Dynamics 44
3.2.5 Multilevel Dynamics 46
3.3 Finite Element Implementation 49
3.3.1 Data Structures 49
3.3.2 Normal Elements 51
3.3.3 Special Elements 51
vi

3.3.4 Force Application 52
3.3.5 Discrete Dynamic Equation 54
3.3.6 Model Subdivision 55
3.4 Generalization of the Approach 56
4 DYNAMIC BUTTERFLY SUBDIVISION SURFACES 57
4.1 Overview of the (Modified) Butterfly Subdivision Scheme 57
4.2 Formulation 61
4.2.1 Local Parameterization 62
4.2.2 Dynamics 72
4.2.3 Multilevel Dynamics 74
4.3 Finite Element Implementation 76
4.3.1 Elemental Mass and Damping matrices 77
4.3.2 Elemental Stiffness Matrix 79
4.3.3 Other Implementation Issues 81
4.4 Generalization of the Approach 82
5 UNIFIED DYNAMIC FRAMEWORK FOR SUBDIVISION SURFACES . 83
5.1 Overview of the Unified Approach 83
5.2 Unified Dynamic Framework for Catmull-Clark Subdivision Surfaces . 85
5.2.1 Local Parameterization 88
5.2.2 Dynamics 92
5.2.3 Finite Element Implementation 92
5.3 Unified Dynamic Framework for Loop Subdivision Surfaces 95
5.3.1 Local Parameterization 97
5.3.2 Dynamics 100
5.3.3 Finite Element Implementation 100
5.4 A General Outline of the Framework for Approximating Subdivision
Schemes 101
5.5 A General Outline of the Framework for Interpolatory Subdivision
Schemes 102
6 MULTIRESOLUTION DYNAMICS 103
6.1 Overview of Multiresolution Analysis and Wavelets 109
6.2 Multiresolution Analysis for Surfaces of Arbitrary Topology 114
6.2.1 Nested Spaces through Subdivision 116
6.2.2 Inner Product 118
6.2.3 Biorthogonal Surface Wavelets on Arbitrary Manifold 119
6.3 Multiresolution Representation 125
6.4 Dynamics 128
6.5 Implementation Details 129
vii

7 SYSTEM ARCHITECTURE 133
7.1 Topological Information Processing Module 135
7.2 Subdivision Module 135
7.3 Finite Element Analysis Module 135
7.4 Force Synthesis Module 136
7.5 Update Engine 136
7.6 Display Module 137
8 APPLICATIONS 138
8.1 Geometric Modeling 138
8.2 Shape Recovery from Range and Volume Data 142
8.3 Non-rigid Motion Estimation 148
8.4 Multiresolution Editing 150
8.5 Natural Terrain Modeling 151
9 CONCLUSIONS AND FUTURE WORK 155
9.1 Conclusions 155
9.2 Future Directions 156
9.2.1 Automatic Change of Topology 156
9.2.2 Local Refinement 157
9.2.3 Automatic Model Parameter Selection 157
9.2.4 Constraint Imposition 157
9.2.5 Recovery of Sharp Features 158
9.2.6 Automatic Model Initialization 158
9.2.7 Improved Synthesized Force Fields 158
REFERENCES 159
BIOGRAPHICAL SKETCH 168
viii

LIST OF FIGURES
2.1 Refinement of an initial control mesh to obtain the limit surface.
(Courtesy : H. Hoppe) 12
3.1 Catmull-Clark subdivision : (a) initial mesh, (b) mesh obtained after
one step of Catmull-Clark subdivision, and (c) mesh obtained after
another subdivision step 28
3.2 A rectangular mesh and its limit surface consisting of 4 bicubic surface
patches 32
3.3 A mesh with an extraordinary point of degree 3 and its limit surface. 34
3.4 Local subdivision around the extraordinary point and the limit surface. 35
3.5 Local subdivision around the extraordinary point and the correspond¬
ing patches in the limit surface from different levels of subdivision. . . 43
3.6 The data structure used for dynamic subdivision surface implementation. 50
3.7 Model subdivision to increase the degrees of freedom : (a) evolution
of the initial mesh, and (b) the corresponding limit surface evolution
perceived by the user 55
4.1 (a) The control polygon with triangular faces; (b) mesh obtained after
one subdivision step using butterfly subdivision rules 58
4.2 (a) The contributing vertices in the j-th level for the vertex in the j-l-l-
th level corresponding to the edge between v( and v^; (b) the weighing
factors for different vertices 58
4.3 (a) The weighing factors of contributing vertex positions for an edge
connecting two vertices of degree 6; (b) the corresponding case when
one vertex is of degree n and the other is of degree 6 60
4.4 The smoothing effect of the subdivision process on the triangles of the
initial mesh 62
IX

4.5 Tracking a point x through various levels of subdivision : (a) initial
mesh, (b) the selected section (enclosed by dotted lines) of the mesh
in (a), after one subdivision step, (c) the selected section of the mesh
in (b), after another subdivision step 64
4.6 Different set of control points at a subdivided level obtained by apply¬
ing different subdivision matrices on a given set of control points in a
coarser mesh 67-68
4.7 (a) An initial mesh, and (b) the corresponding limit surface. The do¬
mains of the shaded elements in the limit surface are the corresponding
triangular faces in the initial mesh. The encircled vertices in (a) are
the degrees of freedom for the corresponding element 75
5.1 A control mesh with an extraordinary vertex of degree 5 and the corre¬
sponding limit surface : (a) using the concepts developed in Chapter 3,
where the limit surface consists of quadrilateral normal elements and
a pentagonal special element; (b) using the unified concept developed
in this chapter, where the limit surface consists of one single type of
quadrilateral finite element 86
5.2 In Catmull-Clark subdivision, each non-boundary quadrilateral face in
the control mesh has a corresponding quadrilateral patch in the limit
surface : (a) control mesh, (b) limit surface 87
5.3 (a) The marked 16 control vertices define the shaded quadrilateral
patch associated with the shaded regular face in the control mesh; (b)
the marked 14 control vertices define the shaded quadrilateral patch
associated with the shaded irregular facie in the control mesh 91
5.4 (a) The control polygon with triangular faeces; (b) mesh obtained after
one subdivision step using Loop’s subdivision rules 96
5.5 An initial mesh and the corresponding limit surface obtained using
Loop’s subdivision rules. The domains of the shaded triangular patches
in the limit surface are the corresponding triangular faces in the initial
mesh. The encircled vertices are the control vertices for the corre¬
sponding triangular patch in the limit surface 98
5.6 Each triangular patch in the limit surface can be associated with a
non-boundary triangular face in the initial mesh, which in turn can be
parameterized over a triangle with vertices at (0, 0), (1, 0) and (0,1). . 99
6.1 Schematic block diagram of the multilevel dynamic's approach 104
x

6.2 Multilevel dynamics vs. multiresolution dynamics 106
6.3 Representation of the degrees of freedom in multilevel dynamic's and
multiresolution dynamic's approach 108
6.4 The filter bank 112
6.5 Subdivision refinement of a tetrahedron 119
6.6 Wavelet construction 123
7.1 System architecture 134
8.1 (a), (b), (e) and (d) : Initial shapes (obtained applying Catmull-Clark
subdivision rules on control meshes); (e), (f), (g) and (h) : the corre¬
sponding modified shapes after interactive force application 139
8.2 (a), (b), (c) and (d) : Initial shapes (obtained applying modified but¬
terfly subdivision rules on control meshes); (e). (f), (g) and (h) : the
corresponding modified shapes after interactive sculpting via force ap¬
plication 141
8.3 (a) Range data of a bulb along with the initialized model, (b) an
intermediate stage of evolution, and (e) the fitted dynamic.' Catmull-
Clark subdivision surface model 143
8.4 (a) Range data of an anvil along with the initialized model, (b) an
intermediate stage of evolution, and (e) the fitted dynamic.' Catmull-
Clark subdivision surface model 144
8.5 (a) Range data of a head along with the initialized model, (d) the
fitted dynamic butterfly subdivision model, and (c) visualization of
the shape from another view point 144
8.6 (a) A slice from a brain MRI, (b) initialized model inside the region
of interest superimposed on the slice, (e) the fitted model superim¬
posed on the slice, and (d) a 3D view of the dynamic.' Catmull-Clark
subdivision surface model fitted to the cerebellum 146
8.7 (a) Data points identifying the boundary of the caudate nucleus on
a MRI slice of human brain, (b) data points (from all slices) in 3D
along with the initialized model, and (e) the fitted dynamic.' butterfly
subdivision model 146
xi

8.8 Snapshots from the animation of canino heart motion over a cardiac
cycle using the dynamic butterfly subdivision model 149
8.9 (a) The initialized model along with the data set; (b) the fitted model
with two subdivisions on the initial mesh, along with attached springs
for editing. The model after editing (c) at lower resolution, (d) at the
same resolution of the fitted model, and (e) at higher resolution. . . . 150
8.10 (a) Discrete elevation data set (4096 points), (b) fitted dynamic but¬
terfly subdivision surface model with 841 vertices (without noise ad¬
dition), and (c) fitted dynamic subdivision surface model with 841
vertices when Gaussian noise is added 153
8.11 Synthesized natural terrain of different roughness using the dynamic
butterfly subdivision surface model with 841 vertices from a data set
of 3099 elevation values 154
xii

Abstract of Dissertation
Presented to the Graduate School of the University of Florida
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
A DYNAMIC FRAMEWORK FOR SUBDIVISION SURFACES
By
Chhandomay Mandal
December 1998
Chairman: Dr. Baba C. Vemuri
Cochairman : Dr. Hong Qin
Major Department: Computer and Information Science and Engineering
Subdivision surfaces are extensively used to model smooth shapes of arbitrary
topology. Recursive subdivision on a user-defined initial control mesh generates a
visually pleasing smooth surface in the limit. However, users have to carefully select
the initial mesh and/or manipulate the control vertex positions at different levels
of subdivision hierarchy to satisfy the functional and aesthetic requirements in the
smooth limit surface. This modeling drawback results from the lack of direct manip¬
ulation tools for the limit surface. In this dissertation, techniques from physics-based
modeling are integrated with geometric subdivision methodology, and a dynamic
framework is presented for direct manipulation of the smooth limit surface generated
by the subdivision schemes using physics-based “force” tools.
xiii

In a typical subdivision scheme, the user starts with an initial control mesh
which is refined recursively using a fixed set of subdivision rules, and a smooth sur¬
face is produced in the limit. Most often this limit surface does not have an analytic.'
expression, and hence poses challenging problems in incorporating mass and damp¬
ing distribution functions, internal deformation energy, forces, and other physical
quantities required to develop a physics-based subdivision surface model. In this
dissertation, local parameterization techniques suitable for embedding the geometric.'
subdivision surface model in a physics-based modeling framework have been devel¬
oped. Specific.' local parameterization techniques have been fully developed for the
Catmull-Clark, modified butterfly and the Loop subdivision schemes. Techniques
for assigning material properties to geometric.' subdivision surfaces are derived, and
a motion equation for the dynamic.' model has been formulated using Lagrangian
dynamic's. Furthermore, advantages of the physics-based deformable models are in¬
corporated into the conventional subdivision schemes, and a dynamic.' hierarchical
control of this model is introduced. Finally, a multiresolution representation of the
control mesh is developed and a unified approach for deriving subdivision surface-
based finite elements is presented.
The proposed dynamic.' framework enhances the applicability of the subdivision
surfaces in modeling applications. It is also useful for hierarchical shape recovery
from large range and volume data sets, as well as for non-rigid motion tracking from
a temporal sequence of data sets. Multiresolution representation of the initial mesh
controlling the smooth limit surface enables global and local editing of the shape as
xiv

desired by the modeler. This dynamic framework has also been used for synthesizing
natural terrain models from sparse elevation data.
xv

CHAPTER 1
INTRODUCTION
Generating smooth surfaces of arbitrary topology poses a grand challenge for
the computer graphic's, computer-aided geometric.1 design and scientific.1 visualization
researchers. The existing techniques for modeling smooth complex shapes can be
broadly classified into two distinct categories namely, (1) explicit patching using
parametric surfaces and (2) subdivision surfaces.
The explicit patching technique involves partitioning the model surface into a
collection of individual parametric surface patches. Adjacent surface patches are
then explicitly stitched together using continuity constraints. This explicit stitching
process is very complicated for modeling smooth surfaces of arbitrary topology due
to the continuity constraints which need to be satisfied along the patch boundaries.
Subdivision surfaces are simple procedural models whic.ii offer an alternate rep¬
resentation for the smooth surfaces of arbitrary topology. A typical recursive subdi¬
vision scheme produces a visually pleasing smooth surface in the limit by repeated
application of a fixed set of refinement rules on an user-defined initial control mesh.
The initial control mesh is a simple polygonal mesh of the same topological type
as that of the smooth surface to be modeled. At each step of subdivision, a finer
polygonal mesh with more vertices and faces is constructed from the previous one via
1

2
the refinement process, and the smooth surface is obtained in the limit. However, a
few subdivision steps on the initial mesh generally suffice to approximate the smooth
surface for all practical purposes. Various sets of rules lead to different subdivision
schemes which mainly differ in the smoothness property of the resulting limit surface
and/or the type of initial mesh (i.e. triangular, quadrilateral etc.) chosen.
1.1 Problem Statement
To model a smooth surface of arbitrary topology using subdivision surfaces, first
a polygonal mesh of same topology needs to be chosen as the initial mesh. This ini¬
tial mesh, also known as the control mesh, is refined via recursive subdivision using
a fixed set of rules, and a smooth surface of the desired topology is obtained in the
limit. However, users have to carefully select the initial mesh and/or manipulate
the control vertex positions at different levels of subdivision hierarchy in order to
satisfy the functional and aesthetic requirements on the smooth limit surface. For
example, to obtain a desired effect on the smooth limit surface, it might be necessary
to reposition a handful of vertices in the mesh obtained after one subdivision step,
or a large number of vertices might need to be moved in the mesh produced after
three subdivision steps! This process is not intuitive and at best laborious. Despite
the presencie of a variety of subdivision schemes in the computer graphic's and geo¬
metric.1 modeling literature, there is no direct and natural way of manipulating the
limit surface. The current state-of-the-art only permits the modeler to interactively
obtain the desired effects on the smooth surface by kinematically manipulating the
vertex positions at various levels of the subdivision hierarchy. In this dissertation, the

3
challenging problem of directly manipulating the smooth limit surface at arbitrary
locations/areas is addressed and a novel solution is presented where the modeler can
not only manipulate the smooth limit surface directly but also control the extent of
the manipulation effect (i.e.. global or local manipulation) on the limit surface.
Subdivision surfaces have also been used for recovering shapes from a given set of
points in 3D. However, most of the existing subdivision surface-based shape recovery
techniques resort to complex algorithms to derive a mesh for the underlying shape,
and then mesh optimization techniques are used to obtain a compact representation
of the same. Nevertheless, this process yields a control mesh of the smooth subdi¬
vision surface representing the underlying shape that typically uses a large number
of degrees of freedom (control vertices) for representation. In this dissertation, an
efficient hierarchical method of recovering shapes from range and volume data sets is
proposed where the control mesh of the smooth limit surface will use very few degrees
of freedom for representation.
1.2 Proposed Solution
Physics-based modeling techniques offer a potential solution to the problem
of directly manipulating the smooth limit surface generated by the recursive sub¬
division procedure. In the physics-based modeling paradigm, a deformable model
is derived by assigning mass, damping, stiffness and other physical properties to a
surface model. The model is deformed by applying synthesized forces and this de¬
formation is governed by physical laws. Xow, if the purely geometric.1 subdivision
surface models can be embedded in a physics-based modeling paradigm, then the

4
smooth limit surface can be deformed directly by using physics-based “force” tools.
However, this procedure-based surface model obtained through the subdivision pro¬
cess does not have a closed-form analytic formulation in general, and hence poses
challenging problems to incorporate mass and damping distribution functions, inter¬
nal deformation energy, forces, and other physical quantities required to develop a
physics-based model. Techniques of locally parameterizing the smooth limit surface
generated by various subdivision schemes are proposed in this dissertation. Once a
suitable local parameterization scheme is developed for a specific subdivision scheme,
a dynamic framework is provided for the corresponding subdivision scheme where
the modeler can directly manipulate the smooth limit surface by using synthesized
forces. At the same time, a dynamic framework for the wavelets derived using subdi¬
vision schemes will assist in adopting a multiresolution representation of the control
mesh defining the smooth limit surface, and the modeler can control the extent of
manipulation effect by choosing a desired level of editing. Synthesized force applica¬
tion at a lower resolution will yield a global effect whereas manipulations at a finer
resolution will have localized effects on the limit surface. The motion of this physics-
based deformable subdivision surface model is governed by a second-order differential
equation, which is solved numerically using the finite element method. Xew types of
finite elements for the chosen subdivision scheme are also presented for representing
the smooth limit surface.
The proposed dynamic framework for the subdivision surfaces provides an effi¬
cient solution to the shape recovery problem as well. A simple subdivision surface

5
model with very few vertices in the control mesh can be initialized (positioned) fully
inside a given set of points in 3D. The initialized model will be deformed by applying
forces synthesized from the given data points. When an equilibrium is obtained, the
number of vertices in the control mesh can be increased via a subdivision step on the
current control mesh thereby increasing the degrees of freedom for model representa¬
tion, and a new equilibrium with a better fit to the given data set can be obtained.
This process can be repeated till a prescribed error in fit is achieved. Similar approach
can be taken for shape recovery from volume data sets as well where a different type
of synthesized force needs to be specified. The hierarchical shape recovery process
ensures a compact representation of the recovered smooth limit surface using very
few degrees of freedom.
1.3 Contributions
In this dissertation, techniques from physics-based modeling are integrated with
geometric.1 subdivision methodology to present a scheme for directly manipulating
the smooth limit surface generated by the subdivision process. As a result, unlike
the existing geometric.' solutions that only allow the operations on control vertices,
the proposed methodology and algorithms permit the user to physically modify the
shape of subdivision surfaces at desired locations via application of forces. This gives
the user a “virtual” clay/play-dough modeling environment. The proposed model
can be edited directly in a hierarchical fashion using synthesized forces. Also, this
physics-based subdivision surface model efficiently recovers shapes as well as non-
rigid motions from large range and volumetric.' data sets. Note that this dissertation

6
neither proposes a new subdivision technique nor provides a different interpretation
of any existing subdivision technique, but integrates the advantages of subdivision
surface-based and physics-based modeling techniques to solve important theoretical
and practical problems. Although the principles of physics-based modeling are well
understood by the graphic's experts and modeling researchers, this dissertation will
greatly advance the state of the art in physics-based shape modeling due to the
contributions listed below.
• Local parameterization techniques for the smooth limit surface generated by
various subdivision schemes are systematically derived in a hierarchical frame¬
work, and subsequently the initial control polyhedron can be viewed as the
parametric domain of the physics-based smooth limit surface.
• The smooth dynamic.1 subdivision surface in the limit is treated as a “real”
physical object represented by a set of novel finite elements. The basis (shape)
functions of these new variety of finite elements are derived using the subdivision
schemes. The proposed finite element methods will prove to be useful not only
in the areas of computer graphic's and geometric.1 design, but also in engineering
analysis as well.
• The subdivision techniques are used to create a surface model that incorporates
mass and damping distribution functions, internal deformation energy, forces,
and other physical quantities. The motion equations are also systematically
derived for this dynamic.1 subdivision surface model whose degrees of freedom are
the collection of initial user-specified control vertices. Therefore, the advantages

7
of both the physics-based modeling philosophy and the geometric subdivision
schemes are incorporated within a single unified framework.
• Users will be able to manipulate this physics-based model in an arbitrary region,
and the model will respond naturally (just like a real-world object would) to
this force application. This shape deformation is quantitatively characterized
by the time-varying displacement of control points that uniquely define the
geometry of the limit surface.
• The dynamic.1 framework for wavelets derived using subdivision schemes en¬
ables a multiresolution representation of the control mesh defining the smooth
limit surface. This provides additional flexibility to the physics-based modeling
framework since the modeler is free to choose the desired editing level. If a
lower resolution editing level is chosen, the synthesized force application on the
smooth limit surface will have global effects, whereas editing with force-based
tools at a finer resolution will yield a localized effect.
• Algorithms and procedures are developed which approximate the proposed new
finite elements using a collection of linear and/or bilinear finite elements subject
to the implicit geometric.' constraints enforced by the subdivision rules. This
hierarchically-structured approximation can satisfy any user-specified error tol¬
erance.
The proposed dynamic.' framework enhances the applicability of subdivision sur¬
face models in various application areas. It provides a direct and intuitive way of

8
manipulating shapes in a hierarchical fashion for geometric modeling applications.
It has also been successfully used for efficient and hierarchical shape recovery from
range and volume data sets as well as for tracking a shape of interest from a time
sequence of range or volume data sets. Finally, the dynamic framework of subdivision
surfaces is combined with a variant of a fractal surface synthesis technique to present
a novel natural terrain modeling method.
1.4 Outline of Dissertation
The rest of the dissertation is organized as follows :
Chapter 2 contains an overview of the subdivision surfaces along with a review of
the related literature. The motivation for embedding the subdivision surface models
in a physics-based modeling framework is discussed, and the proposed model is com¬
pared with the existing physics-based models. The advantages of a unified framework
for shape modeling and shape recovery are also pointed out in this chapter.
Chapter 3 provides a dynamic framework for the Catmull-Clark subdivision
scheme, which is one of the most popular subdivision techniques for modeling compli¬
cated objects of arbitrary genus. Analytic formulation of the limit surface generated
by the Catmull-Clark subdivision scheme is derived and the “physical” quantities
required to develop the dynamic model are introduced [57, 60, 76]. The governing
dynamic differential equation is derived using Lagrangian mechanics and is imple¬
mented using a finite element technique.

9
In Chapter 4, a dynamic.1 framework is provided for the butterfly subdivision
scheme, another popular subdivision technique for modeling smooth surfaces of arbi¬
trary topology. A local parameterization scheme for the butterfly subdivision surface
is derived in a hierarchical style. The physics-based butterfly subdivision model is
formulated as a set of novel finite elements which are optimally approximated by
a collection of standard finite elements subjected to implicit geometric.1 constraints
[58, 61].
Chapter 5 presents an unified approach for providing a dynamic.1 framework for
subdivision surfaces in general. In particular, it has been shown that the limit surface
obtained using any subdivision scheme can be viewed as a collection of either quadri¬
lateral or triangular finite elements whose basis (shape) functions can be derived
using the chosen subdivision scheme [59].
In Chapter 6, a dynamic.1 framework is presented for the subdivision surface-
based wavelet schemes. This dynamic.1 framework enables a multiresolution represen¬
tation of the control mesh defining the smooth limit surface and consequently, the
modeler can control the extent of the effect of manipulation on the limit surface by
choosing a proper editing level.
In Chapter 7, a system that integrates the implementation of the concepts pro¬
posed in earlier chapters is presented. Several modules comprise the entire system,
and the functionality of these modules are discussed in this chapter.

10
The proposed modeling framework is used in various application areas and the
results are presented in Chapter 8. The proposed dynamic model has been success¬
fully used in geometric modeling, shape recovery and non-rigid motion estimation
applications. A novel technique for natural terrain modeling is also presented where
the dynamic framework is combined with a variant of a fractal surface synthesis
technique.
Finally, conclusions are drawn in Chapter 9 where future directions of research
are also pointed out.

CHAPTER 2
BACKGROUND
In this chapter, a brief overview of the subdivision surfaces is presented followed
by a review of the previous work done in the area of subdivision surfaces. This is
followed by an overview of the physics-based models along with a review of the re¬
lated literature. The motivating factors for embedding geometric subdivision surface
models in a physics-based modeling paradigm are presented in the next section. The
advantages of the proposed dynamic models over the existing physics-based models
for shape recovery are discussed and finally advantages of a unified framework for
shape modeling and shape recovery are also presented.
2.1 Subdivision Surfaces
The input to any subdivision scheme is an initial mesh (also known as control
mesh), M° = (Vo, F°), which is a collection of vertices Vo and a collection of faces
F°. The subdivision surface S(M°) associated with the initial mesh M° is defined
as the limit of the recursive application of the refinement R as shown in Fig.2.1.
The refinement R, when applied on a mesh Mk = (Vk,Fk), (.'reates a refined mesh
Mk+1 = (Vk+1. Fk+1) where the vertices in Vk+1 are computed as affine combinations
of the vertices in Vk and the faces in Fk+1 are obtained by splitting each face in Fk
into a fixed number of sub-faces.
11

12
Figuro 2.1. Refinement of an initial control mesh to obtain the limit surface. (Cour¬
tesy : H. Hoppe)

13
It may bo noted that the vortices introduced through the subdivision process
have a fixed degree in general (4 in case of quadrilateral meshes and 6 in case of tri¬
angular meshes). The vertices which do not have this degree are called extraordinary
vertices. The number of extraordinary vertices are very few as those vertices are not
introduced via subdivision refinement in general. For example, in the Catmull-Clark
subdivision scheme (defined on quadrilateral meshes) the number of extraordinary
vertices does not change after the first subdivision on the initial mesh, whereas in the
modified butterfly subdivision scheme (defined on triangular meshes) the subdivision
process never introducios an extraordinary vortex. The limit surface defined by the
subdivision process is most often C1 (first derivative continuous) or C2 (second deriva¬
tive continuous) depending on the subdivision rules expect at very few extra,ordinary
points corresponding to the extraordinary vertices in the mesh. Specific subdivision
schemes along with the properties of their limit surfaces will bo discussed in later
chapters.
A lot of research have boon carried out on subdivision surfaces, which are mainly
focused either on the development of a now subdivision technique or on analyzing
the smoothness properties of the limit surface generated by a subdivision scheme.
However, in this dissertation the focus is entirely different, namely, embedding the
subdivision surfaces in a physics-based modeling paradigm to enhance the applicabil¬
ity of those surface models. In the rest of this section, those “traditional” techniques
of subdivision surfaces are reviewed.

14
Chaikin [14] first introduced the concept of subdivision to the computer graphic's
community for generating a smooth curve from a given control polygon. During the
last two decades, a wide variety of subdivision schemes for modeling smooth surfaces
of arbitrary topology have been derived following Chaikin’s pioneering work on curve
generation. In general, these subdivision schemes can be categorized into two distinct
classes namely, (1) approximating subdivision techniques and (2) interpolator^- sub¬
division techniques. The limit surface obtained using an approximating subdivision
technique only approximates the initial mesh, whereas the limit surface interpolates
the initial mesh in case of interpolator^- subdivision techniques.
Among the approximating schemes, the techniques of Doo and Sabin [27, 83]
generalize the idea of obtaining uniform biquadratic.1 patches from a rectangular con¬
trol mesh. This scheme is an example of vertex subdivision scheme where a vertex
surrounded by k faces is split into k sub-vertices, one for each face. Doo and Sabin’s
subdivision technique can be applied to any mesh of arbitrary topology, and the re¬
sulting smooth limit surface would be biquadratic B-splines, except at extraordinary
points corresponding to the extraordinary vertices in the control mesh.
Catmull and Clark [10] developed a method for recursively generating a smooth
surface from a polyhedral mesh of arbitrary topology. The Catmull-Clark subdivision
surface, defined by an arbitrary initial mesh, can be reduced to a set of standard
bicubic B-spline patches except at a finite number of degenerate points. This scheme
is an example of face subdivision scheme, where a A’-sided face is split into k sub¬
faces. The details of this subdivision scheme are discussed in Section 3.1. Halstead

15
ot al. [38] proposed an algorithm to construct a Catmull-Clark subdivision surface
that interpolates the vertices of a mesh of arbitrary topology.
Loop [51] presented a subdivision scheme based on the generalization of quartic
triangular B-splines for triangular meshes. Loop’s subdivision rules are presented in
Section 5.3. Hoppe ot al. [40] extended Loop’s work to produce piecewise smooth
surfaces with discontinuities in selected areas of the limit surface. They introduced
now subdivision rules that allow for sharp features such as creases and corners in the
limit surface.
Peters and Roif [73] proposed a simple subdivision scheme for smoothing poly-
hodra. Their refinement rules yield a C1 surface that has a piecewise quadratic.1 pa¬
rameterization except at isolated extraordinary points. Most recently, non-uniform
Doo-Sabin and Catmull-Clark surfaces that generalize non-uniform tensor product
B-splinc surfaces to arbitrary topologies wore introduced by Sederberg ot al. [88]. All
the schemes mentioned above generalize recursive subdivision schemes for generating
limit surfaces with a known parameterization. Various issues involved with character
animation using those approximating subdivision schemes wore discussed at length
by DoRoso ot al. [25].
The most well-known interpolation-based subdivision scheme is the “butterfly”
algorithm proposed by Dyn ot al. [30]. Butterfly subdivision method, like other sub¬
division schemes, makes use of a small number of neighboring vortices for subdivision.
It requires simple data structures and is extremely easy to implement. However, it
needs a topologically regular sotting of the initial (control) mesh in order to obtain

16
a smooth C1 limit surface. A variant of this scheme with better smoothness prop¬
erties can be found in Dyn et al. [29]. Zorin et al. [109] developed an improved
interpolatory subdivision scheme (which we (.'all the modified, butterfly scheme) that
retains the simplicity of the butterfly scheme and results in much smoother surfaces
even from irregular initial meshes. These interpolatory subdivision schemes have
extensive applications in wavelets on manifolds, multiresolution decomposition of
polyhedral surfaces and multiresolution editing.
A variational approach for interpolatory refinement has been proposed by Kobbelt
[43, 44] and by Kobbelt and Schroder [46]. In this approach, the vertex positions in
the refined mesh at each subdivision step are obtained by solving an optimization
problem. Therefore, these schemes are global, i.o., every new vertex position depends
on all the vertex positions of the coarser level mesh. The local refinement property
which makes the subdivision schemes attractive for implementation in the computer
graphic's applications is not retained in the variational approach.
The derivation of various mathematical properties of the smooth limit surface
generated by the subdivision algorithms is rather complex. Doo and Sabin [28] first
analyzed the smoothness behavior of the limit surface using the Fourier transform and
an eigen-analysis of the subdivision matrix. Ball and Stony [3, 4] and Reif [80] further
extended Doo and Sabin’s prior work on continuity properties of subdivision surfaces
by deriving various necessary and sufficient conditions on smoothness for different
subdivision schemes. Specific.1 subdivision schemes were analyzed by Schweitzer [87],
Habib and Warren [37], Peters and Reif [74] and Zorin [108]. Most recently, Stam

17
[90] presented a method for exact evaluation of Catmull-Clark subdivision surfaces
at arbitrary parameter values. It may be noted that the focus of this dissertation
is not on deriving a new subdivision technique or analyzing a subdivision technique,
but on deriving a deformable surface model using these subdivision schemes.
2.2 Physics-based Deformable Surface Models
Shape models can be broadly categorized into two types - lumped parameter
models and distributed parameter models. A lumped parameter model uses small
number of parameters to represent model geometry, whereas a distributed parameter
model uses large number of degrees of freedom for model representation. An example
of a lumped parameter surface model is a superquadric. Spline surfaces serve as a
good example for distributed parameter model.
Deformable surfaces are distributed parameter models which use large number
of degrees of freedom for representing the model geometry. A large variety of shapes
can be modeled using this type of model, but handling a large number of degrees of
freedom can be cumbersome. However, the large degrees of freedom of a deformable
model are embedded in a physics-based framework to allow only a “physically mean¬
ingful” model behavior. Various types of energies are assigned to the model using the
degrees of freedom, and the model “deforms” to find an equilibrium state with min¬
imal energy. The motion equation is derived using Lagrangian dynamic's [36], where
various energies associated with model gives rise to internal and external forces. The
equilibrium state of the model is a model position where the internal deformation
force becomes equal to the externally applied force. These physics-based deformable

18
models are useful for modeling where the modeler can deform a surface by applying
synthesized forces, and for data fitting where external forces are synthesized from a
given data set such that the model approximates the given data at equilibrium.
A deformable surface model is typically parameterized over the domain [0, l]2.
Let s(u, v, p) be a deformable surface model, where 0 < u, v < 1 and p is a collection
of the degrees of freedom Pi, i = 1,2,, n associated with the model (assuming the
model has n degrees of freedom). The degrees of freedom p,¿(í) is a set of generalized
coordinates which are functions of time and are assembled into the vector p. Let
fi(t) be the generalized force represented by the vector fp and acting on p,¿. Let T be
the kinetic.1 energy, F be the dissipation energy and U be the potential energy of the
deformable surface model. Then, the Lagrangian equation of motion for the model
can be expressed as
d_0T_ _ dT (W
dt dpi dpi + dpi + dp,¿
(2.1)
Let /./,(«, v) be the mass density function of the surface. Then the kinetic.' energy
of the surface is
T
1
2
f.i s1 sdudv
(2.2)
whore M is called the mass matrix. Similarly, let 7(u.v) be the damping density
function of the surface. Then the dissipation energy is
^PiDP,
F
1
2
7s7 sdudv
(2.3)

19
where D is (.'ailed the damping matrix. The potential energy of the model can be
defined using the tMn-plate-under-tension energy model [96], and is given by
U
Qill
+ /hi
dsT ds
d u d u
d2sT d2s
dudodudo
+ «22-
dsT Os
do do
+ (h
d2sT o2i
d2u 02u
+ p22
d2sT d2s
d2o d2v
)dudo
lpJKp,
(2.4)
where <\¿¿( respectively of the deformable surface model, and K is known as the stiffness matrix.
The discretized equation of motion can be derived using the expressions of the kinetic.',
dissipation and potential energy in Eqn.2.1, which is given by
Mp + Dp + Kp = fp, (2.5)
where fp is the generalized force vector.
The free-form deformable models discussed above were first introduced to com¬
puter graphic's and visualization in Terzopoulos et al. [99] and further developed by
Terzopoulos and Fleischer [97], Pentland and Williams [72], Metaxas and Terzopou¬
los [65] and Vemuri and Radisavljevic [104], Celniker and Gossard [11] developed a
system for interactive free-form design based on the finite element optimization of en¬
ergy functionals proposed in Terzopoulos and Fleischer [97]. Bloor and Wilson [8, 9],
Celniker and Welch [12] and Welch and Witkin [105] proposed deformable B-spline
curves and surfaces which can be designed by imposing the shape criteria via the
minimization of the energy functionals subject to hard or soft geometric.' constraints

20
through Lagrange multipliers or penalty methods. Qin and Terzopoulos [77, 78, 100]
developed dynamic XL7RBS (D-XTJRBS) which are very sophisticated models suit¬
able for representing a wide variety of free-form as well as standard analytic shapes.
The D-XXRBS have the advantage of interactive and direct manipulation of XXRBS
curves and surfaces, resulting in physically meaningful hence intuitively predictable
motion and shape variation.
Deformable models are also widely used for shape recovery, segmentation, mo¬
tion tracking and other computer vision and medical imaging applications. A detailed
survey of deformable models used in these techniques can be found in Mclnerney and
Terzopoulos [64] and the references therein. Some specific existing deformable models
used for shape recovery and non-rigid motion estimation will be reviewed in Section
2.4.
2.3 Shape Modeling Using Physics-based Subdivision-surface Model
Recursive subdivision surfaces are powerful for representing smooth geometric
shapes of arbitrary topology. However, they constitute a purely geometric represen¬
tation, and furthermore, conventional geometric modeling with subdivision surfaces
may be difficult for representing highly complicated objects. For example, modelers
are faced with the tedium of indirect shape modification and refinement through time-
consuming operations on a large number of (most often irregular) control vertices
when using typical subdivision surface-based modeling schemes. Despite the advent
of advanced 3D graphic's interaction tools, these indirect geometric.1 operations remain
non-intuitive and laborious in general. In addition, it may not be enough to obtain

21
the most “fair” surface that interpolates a set of (ordered or unorganized) data points.
A certain number of local features such as bulges or inflections may be strongly de¬
sired while requiring geometric objects to satisfy global smoothness constraints in
geometric modeling and computer graphic's applications. In contrast, physics-based
modeling provides a superior approach to shape modeling that can overcome most of
the limitations associated with traditional geometric.1 modeling approaches. Free-form
deformable models governed by the laws of continuum mechanics are of particular
interest in this context. These dynamic.1 models respond to externally applied forces
in a very intuitive manner. The dynamic.1 formulation marries the model geometry
with time, mass, damping and constraints via a force balance equation. Dynamic.1
models produce smooth, natural motions which are easy to control. In addition, they
facilitate interaction - especially direct manipulation of complex geometries. Fur¬
thermore, the equilibrium state of the model is characterized by a minimum of the
deformation energy of the model subject to the imposed constraints. The deformation
energy functionals can be formulated to satisfy local and global modeling criteria, and
geometric.1 constraints relevant to shape design can also be imposed. The dynamic.1
approac.ii subsumes all of the aforementioned modeling capabilities in a formulation
whicii grounds everything in real-world physical behavior.
A severe limitation of the existing deformable models, including D-XURBS, is
that they are defined on a rectangular parametric.1 domain. Hence, it can be very
difficult to model surfaces of arbitrary genus using these models. In a recent work,

22
DeRose et al. [25] assigned material properties to control meshes at various subdivi¬
sion levels in order to simulate (doth dynamic's using subdivision surfaces. Note that,
they assign physical properties on the control meshes at various levels of subdivision
and not on the limit surface itself, and hence can not solve the modeling goal we
are trying to achieve. In this dissertation, a dynamic.1 framework is presented for
subdivision surfaces which combines the benefits of subdivision surfaces for modeling
arbitrary topology as well as that of dynamic.1 splines for interactive shape manip¬
ulation by applying synthesized forces. The proposed dynamic.1 framework presents
the modeler a formal mechanism of direct and intuitive manipulation of the smooth
limit surface, as if they were seamlessly sculpting a piece of real-world “(day.” A
dynamic.1 framework for subdivision surface-based wavelets adds flexibility in the pro¬
posed modeling paradigm. It enables a multiresolution representation of the evolving
control mesh defining the smooth limit surface, and the modeler can control the
extent of manipulation effect by choosing a proper editing level. The formulation
and implementation details of these dynamic.1 frameworks are discussed in subsequent
chapters.
2.4 Shape and Motion Estimation Using Physics-based Subdivision-surface Model
The dynamic.1 subdivision surface model has been developed primarily for mod¬
eling arbitrary (known) topology where modelers can directly manipulate the limit
surface by applying synthesized forces in an intuitive fashion. However, another im¬
portant application of the dynamic.1 subdivision surfaces is in non-rigid shape and

23
motion reconstruction/recovery. Accurate shape recovery requires distributed pa¬
rameter models which typically possess a large number of degrees of freedom. On
the other hand, efficient shape representation imposes the requirement of geometry
compression, i.e., models with fewer degrees of freedom. These requirements are
conflicting and numerous researchers have been seeking to strike a balance between
these contradicting requirements [5. 21. 47, 49, 62, 89, 100, 104], Another important
criterion in the design of shape models is that the initialization of the model during
the shape recovery process should not be restricted to globally parameterized input
meshes since it may be infeasible to globally parameterize shapes of arbitrary topol¬
ogy. A physics-based model best satisfying the above mentioned criteria is an ideal
candidate for a solution to the shape recovery problem for obvious reasons.
Deformable models which come in many varieties, have been used to solve this
problem in the physics-based modeling paradigm. These models involve the use of
either fixed size [21, 66, 71, 98, 104] or adaptive size [15, 41, 47, 62, 95, 101] grids.
The models with fixed grid size generally use less number of degrees of freedom for
representation at the cost of accuracy of the recovered shape. On the other hand,
models using adaptive grids generally need large number of degrees of freedom to
recover the shapes accurately. Note that the shapes being recovered from the image
data are smooth in most of the medical applications, i.e. the anatomical structures
even with considerable amount of details have more or less a C1 surface. Under
these circumstances, the finite element approaches as in Cohen and Cohen [21] and
Mclnerney and Terzopoulos [62] need a large number of degrees of freedom for deriving

24
a smooth and accurate representation. In addition, they can not represent shapes
with known arbitrary topology. Moreover, almost all of these schemes require a
globally parameterized mesh as their input which may be infeasible at times.
The proposed model solves the shape recovery problem very effectively as it
can recover shapes from large range and volume data sets using very few degrees
of freedom (control vertices) for its representation and can cope with any arbitrary
input mesh, not necessarily parameterized. The initialized model deforms under the
influence of synthesized forces to fit the data set by minimizing its energy. Once the
approximate shape is recovered, the model is further subdivided automatically and a
better approximation to the input data set is achieved using more degrees of freedom.
The process of subdivision after achieving an approximate fit is continued till a pre¬
scribed error criteria for fitting the data points is achieved. The proposed dynamic
subdivision surface models have also been successfully used in motion tracking and
visualization of a dynamically deforming shape from a time sequence of volumetric
data sets.
2.5 Unified Framework for Shape Recovery and Shape Modeling
Currently, shape recovery and shape modeling are viewed as two distinct areas in
computer graphic's and geometric.1 modeling literature. However, there are potential
benefits if these two can be combined in an unified framework. For example, the
modeler starts from scratch to build a specific model in a typical geometric.1 modeling
scenario. First, a rough shape is modeled and then it is fine-tuned by manipulating
control vertex positions to obtain the desired effects. This turns out a cumbersome

25
process in general. On the other hand, shape recovery using state-of-the-art methods
yield large polygonal meshes which are very difficult to manipulate, especially for
global changes in shape. Most often the resulting meshes from a shape recovery
application are not directly amenable to multiresolution analysis. Computationally
expensive re-meshing techniques are needed to convert these meshes into a specific.1
type of meshes on which multiresolution analysis can be performed. This specific.1 type
of mesh is known as mesh with “subdivision-connectivity”, implying a topologically
equivalent mesh with the same connectivity as of the given mesh can be obtained by
recursive subdivision of a very simple known initial mesh.
The proposed dynamic.1 framework combines shape recovery and shape modeling
in an unified framework where the modeler can scan in 3D points of a prototype
model, recover the shape using the proposed dynamic.1 subdivision surface model, and
edit at any desired resolution using physics-based force tools. Thus, the modeler is re¬
lieved of the burden of both building an initial model and editing a cumbersome huge
polygonal mesh. The shape recovery process starts with a smooth subdivision surface
model which has a simple initial mesh. This model is deformed using synthesized
forces from the given data points and is automatically refined using some pre-defined
error in fit criteria. The initial mesh of the final recovered smooth shape has subdivi¬
sion connectivity in-built, as it is obtained by subdivision refinement, and therefore
no re-meshing is needed for multiresolution analysis. The dynamic.1 framework for
subdivision surface-based wavelets makes multiresolution editing using physics-based

26
force tools easy to perform. These advantages of shape modeling and shape recovery
in a unified framework will be further illustrated in later chapters.

CHAPTER 3
DYNAMIC CATMULL-CLARK SUBDIVISION SURFACES
Subdivision surfaces, as mentioned earlier, can be broadly classified into two
categories - approximating and interpolatory subdivision schemes. The approxi¬
mating subdivision schemes typically generalize recursive subdivision techniques for
generating limit surfaces with known parameterizations. In this chapter, a dynamic.'
framework will be presented for Catmull-Clark subdivision scheme, a popular approx¬
imating subdivision technique. The Catmull-Clark subdivision scheme generalizes the
idea of obtaining a bicubic B-spline surface by recursive subdivision of a rectangu¬
lar initial mesh. Before discussing the dynamic.' framework, first the Catmull-Clark
subdivision scheme is briefly reviewed.
3.1 Overview of the Catmull-Clark Subdivision Scheme
Catmull-Clark subdivision scheme, like any other subdivision scheme, starts
with a user-defined mesh of arbitrary topology. It refines the initial mesh by adding
new vertices, edges and faces with each step of subdivision following a fixed set of
subdivision rules. In the limit, a recursively refined polyhedral mesh will converge to
a smooth surface. The subdivision rules are as follows:
27

28
Figuro 3.1. Catmull-Clark subdivision : (a) initial mesh, (b) mesh obtained after one
stop of Catmull-Clark subdivision, and (e) mesh obtained after another subdivision
stop.
• For each face, a now face vortex is introduced which is the average of all the
old vertices defining the face.
• For each (non-boundary) edge, a now edge vortex is introduced which is the
average of the following four points: two old vortices defining the edge and two
now face vortices of the faces adjacent to the edge.
• For each (non-boundary) vortex V, a now vortex is introduced whoso position
is n + ^ ■ wh°r0 F is the average of the now face vortices of all faces
adjacent to the old vortex V, E is the average of the midpoints of all the edges
incident on the old vortex V and n is the number of the edges incident on the
vortex.
• Xow edges are formed by connecting each now face vortex to the now edge
vortices of the edges defining the old face and by connecting each now vortex

29
corresponding to an old vortex to the now edge vertices of all old edges incident
on the old vortex.
• Xow faces are defined as faces enclosed by now edges.
An example of Catmull-Clark subdivision on an initial mesh is shown in Fig.3.1.
The most important property of the Catmull-Clark subdivision surfaces is that a
smooth surface can be generated from any control mesh of arbitrary topology. There¬
fore, this subdivision scheme is extremely valuable for modeling various complicated
geometric objects of arbitrary topology. Catmull-Clark subdivision surfaces include
standard bicubic B-spline surfaces as their special ease (i.e., the limit surface is a
bicubic B-spline surface for a rectangular mesh with all non-boundary vertices of
degree 4). In addition, the aforementioned subdivision rules generalize the recur¬
sive bicubic B-spline patch subdivision algorithm. For non-rectangular meshes, the
limit surface converges to a bicubic B-spline surface except at a finite number of ex¬
traordinary points. These extraordinary points correspond to extraordinary vertices
(vertices whose degree is not equal to 4) in the mesh. Note that, after the first sub¬
division, all faces are quadrilaterals, hence all the new vertices created subsequently
will have four incident edges. The number of extraordinary points on the limit sur¬
face is a constant, and is equal to the number of extraordinary vertices in the refined
mesh obtained after applying one step of the Catmull-Clark subdivision on the initial
mesh. The limit surface is curvature-continuous everywhere except at extraordinary
vertices, where only tangent plane continuity is achieved. In spite of the popularity
of Catmull-Clark subdivision surfaces for representing complex geometric shapes of

30
arbitrary topology, those subdivision surfaces may not bo easily paramotorizablo and
deriving a (dosed-form analytic.1 formulation of the limit surface can be very difficult.
Those deficiencies preclude their immediate pointwisc manipulation and hence may
restrain the applicability of those schemes. In the rest of the chapter, a dynamic.1
framework is developed for the Catmull-Clark subdivision surfaces which offers a
closed-form analytic.1 formulation and allows users to manipulate the model directly
and intuitively. It may bo noted that recently a scheme was proposed by Stam [90]
for directly evaluating the Catmull-Clark subdivision surfaces at arbitrary parameter
values, but the work presented hero on dynamic.1 Catmull-Clark subdivision surfaces
[57, 60, 76] was published prior to the publication of the above-mentioned work.
3.2 Formulation
In this section, a systematic.1 formulation of the now dynamic.1 model based on
Catmull-Clark subdivision surfaces is presented. This dynamic.1 model treats the
smooth limit surface as a function of its initial mesh. However, the control vortex po¬
sitions need to bo updated continually at every level of subdivision in order to develop
the dynamic.1 framework. Note that, all the vertices introduced through subdivision
are obtained as an affine combination of control vortex positions in the initial mesh.
Therefore, the dynamic.1 behavior of the limit surface can bo controlled by formulat¬
ing the dynamic.1 model on the initial mesh itself, the only exception being the (.'aso
when the initial mesh has non-rcctangular faces. This problem can be circumvented
by taking the mesh obtained through one stop of subdivision as the initial mesh.
It may bo noted that an additional subdivision stop may bo needed in some (.'ases

31
to isolate the extraordinary points, and the resulting mesh is treated as the initial
mesh. A typical example of the above mentioned scenario is when the initial mesh is
a tetrahedron where two subdivision steps are needed, and the dynamic framework
can be developed by treating the mesh obtained after two subdivision steps as the
initial mesh. To define the limit surface using the vertices of the initial mesh, the
enumeration of the bicubic patches in the limit surface is necessary. In the next two
sections, various schemes are presented for assigning the bicubic patches to various
faces of the initial mesh.
3.2.1 Assigning Patches to Regular Faces
In Fig.3.2, a rectangular control mesh is shown along with the bicubic B-spline
surface (4 patches) in the limit after an infinite number of subdivision steps. Note
that, each of the bicubic patches in the limit surface is defined by a rectangular face
with each vertex of degree four, thereby accounting for 16 control points (from its
8 connected neighborhood) needed to define a bicubic surface patch in the limit.
Therefore, for each rectangular face in the initial mesh with a degree of 4 at each
vertex, the corresponding bicubic surface patch can be assigned to it in a straight
forward way. In Fig.3.2, the surface patches Si,S2,S^ and 5*4 are assigned to face
Fi, F2, F?> and ip respectively. The 16 control points for the patch Si, corresponding
to face Fi, are highlighted in Fig.3.2. Note that, the initial control mesh can be viewed
as the parametric domain of the limit surface. Therefore, face ip can be thought of
as the portion of the parametric domain over which the patch S\ is defined, i.e., has
non-zero values. Nevertheless, each rectangular face (e.g. ip) can be parametrically

32
Figure 3.2. A rectangular mesh and its limit surface consisting of 4 bicubic surface
patches.

33
defined over [0, l]2, and hence, all bicubic B-spline patches defined by 16 control
points are locally parameterized over [0, l]2.
3.2.2 Assigning Patches to Irregular Faces
In Fig.3.3, a mesh containing an extraordinary point of degree 3 and its limit sur¬
face are shown. The faces F0,Fi, ..., F8 are assigned to bicubic patches S0, Si,..., S%
respectively (as they all have vertices of degree 4) following the aforementioned
scheme. However, the central smooth surface enclosed by the patches S0, Si,..., S%
consists of infinite number of bicubic patches converging to a point in the limit. A re¬
cursive way of enumerating these bicubic patches and assigning them to various faces
at different levels need to be developed in order to formulate the dynamic framework
for Catmull-Clark subdivision surface model.
The idea of enumerating the bicubic patches corresponding to faces having an
extraordinary vertex is shown in Fig.3.4 where a local subdivision of the mesh con¬
sisting of faces Fq, ifi,.... F%, P0, Pi, P2 (and not the other boundary faces) of Fig.3.3
is carried out. Topologically, the resulting local subdivision mesh (shown as dotted
mesh) is exactly the same as the mesh in Fig.3.3 and hence exactly the same number
of bicubic patches can be assigned to its faces with vertices of degree 4 as is evident
from Fig.3.4 (the new faces and the corresponding patches are marked by “p” and
“n” respectively). This process of local subdivision and assignment of bicubic patches
around an extraordinary point can be carried out recursively and in the limit, the
enclosed patch corresponding to faces sharing the extraordinary point will converge
to a point. However, there is no need to carry out an infinite number of subdivision

34
Figuro 3.3. A mesh with an extraordinary point of degree 3 and its limit surface.

35
Figuro 3.4. Local subdivision around the extraordinary point and the limit surface.

36
stops. This description is for formulation purposes only and the exact implementation
will bo detailed in a later section.
3.2.3 Kinematics
In this section, the mathematics for the kinematics of the limit surface is devel¬
oped via illustrative examples and then the generalized formulas are presented. It
may bo noted that a single bicubic B-splinc patch is obtained as the limiting process
of the Catmull-Clark subdivision algorithm applied to an initial 4 by 4 rectangular
control mesh. Let sp(u,v), whore (u.v) e [0, l]2, denote this bicubic B-splinc patch
which can bo expressed analytically as
3 3
sp(u, v) = (.r{ Í/. v), //(ii. v), z(u, v))1 = X X di:jBiA(u)BjA(v), (•■'>• 1)
i=0j=0
whore d*.¿ represents a 3-dimcnsional position vector at the (i.j)th control point
location and BiA(u) and BjA(v) are the cubic B-splinc basis functions. The subscript
p on s denotes the patch under consideration. Expressing Eqn.3.1 in a generalized
coordinate system we have
Sp = JpQp; (3.2)
where 3P is a transformation matrix storing the basis functions of a bicubic B-spline
patch, and is of size (3,48). Vector qp is the concatenation of all control points
defining a B-spline patch in 3D. Note that in the concatenation of the control points,
each control point has an (x, y, z) component. For example, the (x, y, z) components
of the control point (i.j) correspond to positions 3k, 3k + 1,3k + 2 - where, k =

37
4?! + j - respectively in the vector qp. We can express the entries of Jp explicitly
in the following way: Jp(0, k) = Jp(l,k + 1) = Jp(2, A: + 2) = BiA(u)BjA(v) and
Jp(0, k + 1) = Jp(0, k + 2) = Jp(l. k) = Jp(l. k + 2) = Jp(2, k) = Jp(2, k + 1) = 0.
Limit surface with many bicubic patches from a rectangular initial mesh
Xow a limit surface consisting of many bicubic surface patches obtained after
applying an infinite number of subdivision steps to a rectangular initial mesh is
considered. For example, let the limit surface of Fig.3.2 be sm which can be written
as
sm(u, v) = smi(2u. 2v)+sm.2(2(u — -), 2w)+sm3(2(« — -), 2(w^-))+smil(2w,, 2(v — -)),
(3.3)
where smi(2-«, 2?;) = sm(u. v) for 0 < u. v < and 0 otherwise. Similarly, sm2,sma
and smi are also equal to sm(u,v) for an appropriate range of values of u.v and 0
outside. It may be noted that smi, sm2, sm3, smi correspond to patches Si, S2, S3, S4
respectively in Fig.3.2. Eqn.3.3 in generalized coordinates becomes
4
sm = JiQi + J2Q.2 + J.sQs + JrQr = (3.4)
i=l
where J,¿s are the transformation matrices of size (3, 48) and q,¿s are the (x,y,z) com¬
ponent concatenation of a subset of the control points of sm defining sm¿, i = 1, 2, 3

38
and 4. A moro general expression for sm is
4
Sm = JiAiqm + J2A2Qm + J.sAsqm + J4A4qm = 'y ( J,¿A,¿qm = Jmqm. (3-5)
i=l
Whore, qm is the 75-componcnt vector of 3D positions of the 25 vortex control mesh
defining the limit surface sm. Matrices A,¿, 1 < i < 4, are of size (48,75) , each
row consisting of a single nonzero entry (= 1) and the (3, 75)-sizcd matrix Jm =
Eti J,-A,,
Limit surface with many bicubic patches from an arbitrary initial mesh
The stage is now sot to define the limit surface s using the vortices of initial
mesh M for any arbitrary topology, assuming all faces are quadrilateral and no face
contains more than one extraordinary point as its vortex (i.c., extraordinary points
are isolated). As mentioned earlier, if those assumptions are not satisfied, one or two
stops of global subdivision may bo required and the resulting mesh can bo treated
as the initial mesh. Lot the number of vortices in the initial mesh M bo a, and lot
l of those bo the extraordinary vortices. The number of faces in the initial mesh are
assumed to bo b, and k of those faces have vortices with degree 4 (henceforth termed a
“normal face”) and each of the remaining (b — k) faces have one of the l extraordinary
vortices (henceforth termed a “special face”). Lot p bo the 3a = N dimensional vector
containing the control vortex positions in 3D. Using the formulations in Section 3.2.1

39
and Section 3.2.2, the smooth limit surface can be expressed as
k l
¿=1 j=1
(3.6)
where rp is a single bicubic patch assigned to each of the normal faces and Sj is a
collection of infinite number of bicubic patches corresponding to each of the extraor¬
dinary points. As s is a surface defined over the faces of the initial control mesh, each
n* can be viewed as a bicubic patch defined over the corresponding regular face in the
initial control mesh. Similarly, each sj can be defined over the corresponding irregu¬
lar faces in the initial mesh (refer to Fig.3.2 - Fig.3.4 for the detailed description on
parametric domains of subdivision surfaces). For the simplicity of the following math¬
ematical derivations on the dynamic model, from now on the parametric domain will
not be explicitly provided in the formulation. Employing the same approach taken
before to derive Eqn.3.5, it can be shown that
k
k
k
E n, = E("J¡)("p,) = (E("J¡)("A,.))p = ("J)p
(3.7)
i=l i=l i=l
i=l
where "X/," p, and "A,¿ are the equivalent of J,¿, p¿ in Eqn.3.4 and A,¿ in Eqn.3.5 respec¬
tively. The pre-superscript n is used to indicíate that these mathematical quantities
describe bicubic patch in the limit surface corresponding to normal faces.
The following notational convention will be used for describing various math¬
ematical quantities used in the derivation of the expression for a collection of infinite
number of bicubic patches around an extraordinary vertex. The pre-superscript s

40
is used to represent a collection of bicubic patches around an extraordinary vertex,
the subscript j is used to indicate the j-th extraordinary point, the post-superscript
represents the exponent of a mathematical quantity and the level indicator (to rep¬
resent various levels of subdivision in the local control mesh around an extraordinary
vertex) is depicted via a subscript on the curly braces.
The expression for Sj is derived using the recursive nature of local subdivision
around an extraordinary vertex as shown in Section 3.2.2. First, Sj can be expressed
as
sj = + \sj}v (3-8)
where the first term of Eqn.3.8 is the generalized coordinate representation of the
bicubic B-spline patches corresponding to the normal faces of the new local subdivi¬
sion mesh obtained after one subdivision step on the local control mesh (similar to
those patches marked n in Fig.3.4). {sJ}1 represents the rest of the infinite bicubic
B-spline patches surrounding the extraordinary point (similar to the central patch
enclosed by patches marked n in Fig.3.4). The vertices in the newly obtained lo¬
cal subdivision mesh {■‘ipJ}1 can be expressed as a linear combination of a subset
of the vertices of the initial mesh M (which will contribute to the local subdivi¬
sion) following the subdivision rules. We can name this subset of initial control
vertices {•,pj}0. Furthermore, there exists a matrix of size (3c, 3d), such that
fB/lifPjlo = {'P./li where {•,pj}1 and {•,pj}0 are vectors of dimension 3c and 3d
respectively. Applying the idea of recursive local subdivision again on sj can

41
bo further expanded as
sj = fJjlifB/lifPjlo + + {s j}2. (3.9)
In the above derivation, is a vector of dimension 3d, comprising of a
subset of the vortices defining the 3c dimensional vector {■‘ipJ}1. Note that, {■‘ip?}1
has the same structure as {',’Pj}0; therefore, there exists a (3d, 3d) matrix
such that {',’Cj}1{',’Pj}0 = {•,pj}1. Each subdivision of a local mesh with d vortices
(•reatos a now local mesh with c vortices which contributes a fixed number of bicubic
B-splinc patches. Proceeding one stop further, it can bo shown that
SJ = {'-l,}i{j}i{'P./lu + iM'C/11i'P./lu
+ { S }3^ SB4 }3^ S^4 } 1 { SPi }0 + i SJ }3' (3.10)
Because of the intrinsic.1 property of the local recursive subdivision around the ex¬
traordinary point, wo have = ••• = = ••• = In
addition, the subdivision rules remain the same throughout the refinement process,
we also have {*BJ}1 = {SB;}2 = ... = {*Bj}n = ... = {'Bj}^. So, the above
equations can bo further simplified loading to
OO
= {'JUitBUifEt'CUiH'pUr
i=0
(3.11)

42
Vector sj can be written as
CJjJCPj)
(3.12)
where ■‘’ Jj = sCj }') and sp¿ = {',’Pj}0- The idea of local recursive
subdivision around an extraordinary point is illustrated in Fig.3.5. Note that, each
vertex position in the subdivided mesh is obtained by an affine combination of some
vertices in the previous level and hence any row of sums to 1. The largest
eigenvalue of such a matrix is 1 and it can be shown that the corresponding infinite
series is convergent following a similar approach as in Halstead et al. [38]. The rest
of the derivation leading to an expression for s is relatively straight forward. Using
the same approach used to derive the Eqn.3.7, it can be shown that
E s; = B’JiX’pj) = (B’-W-Mp = ('J)p-
(3.13)
j=1 i=1 i=1
j=1
From Eqn.3.6, Eqn.3.7 and Eqn.3.13
s =
(”J)p + (sJ)p.
(3.14)
Let J = ("J) + (SJ). and hence
s = Jp.
(3.15)

43
Figuro 3.5. Local subdivision around the extraordinary point and the corresponding
patches in the limit surface from different levels of subdivision.

44
3.2.4 Dynamics
In the previous section, an expression is derived for the smooth limit surface
obtained via infinite subdivision steps. This expression treats the smooth limit surface
as a function of the control vertex positions in the initial mesh. Now the vertex
positions in the initial mesh defining the smooth limit surface s are treated as a
function of time in order to embed the Catmull-Clark subdivision model in a dynamic
framework. The velocity of this surface model can be expressed as
s(u, v, p) = Jp,
(3.16)
where an overstruck dot denotes a time derivative. The physic's of the dynamic
subdivision surface model is based on the work-energy version of Lagrangian dynamic's
[36] and is formulated in an analogous way to the dynamic.' framework presented in
Section 2.2.
Let fi(u, v) be the mass density function of the surface. Then the kinetic.' energy
of the surface is
T
1
2
//
(3.17)
where (using Eqn.3.16)
(3.18)

45
is the mass matrix of size (N.N). Similarly, lot 7(u.v) bo the damping density
function of the surface. Then the dissipation energy is
F = J'js1 sdudv = ^p^Dp, (3.19)
whore
D = J J 7J2 Jdudo (3.20)
is the damping matrix of size (N, N).
The potential energy of the smooth limit surface can be expressed as
U = ^pJKp, (3.21)
whore K is the stiffness matrix of size (N.N). A fJiin-plate-under-tension energy
model [96] is used to compute the elastic.1 potential energy of the dynamic.1 subdivision
surface. The corresponding expression for the stiffness matrix K is
K = J/(a„JiJ. + o'22^l J-« + (hi^iu^uu + A2J uv^uv + /Í22J lvJVv)dudv. (3.22)
where the subscripts on J demote the parametric.1 partial derivatives. The r)
and [iij(u, ?;)s are elasticity functions controlling local tension and rigidity in the two
parametric.1 coordinate directions. Note that the thin-plato energy expression might
diverge at extraordinary points on the limit surface for Catmull-Clark subdivision
scheme as shown in Halstead et al. [38]. Several methods have been suggested in

46
Halstead ot al. [38] to overcome the problem of the divergent series. However, in this
dissertation, the problem is circumvented by setting the corresponding rigidity coef¬
ficients to be zero at these points. Therefore, the thin-plate energy at extraordinary
points is zero. The effect is negligible to the overall thin-plate energy as very few
extraordinary points are present in the smooth limit surface.
Using the expressions for the kinetic, dissipation and potential energy in Eqn.2.1,
the same motion equation as in Eqn.2.5 can be obtained. The generalized force vector
ip, which (tan be obtained through the principle of virtual work [36], is expressed as
(3.23)
Different types of forces can be applied on the smooth limit surface, and the limit
surface would evolve over time according to Eqn.2.5 to obtain an equilibrium position
characterized by a minimum of the total model energy.
3.2.5 Multilevel Dynamics
At this point, a dynamic framework is developed for the Catmull-Clark subdi¬
vision scheme where the smooth limit surface evolves over time in response to the
applied forces. The entire process can be described as follows. Given an initial mesh,
a smooth surface is obtained in the limit. Users can directly apply synthesized forces
to this smooth limit surface to enforce various functional and aesthetic.1 constraints.
This direct manipulation is then transferred back as virtual forces acting on the initial
mesh through a transformation matrix (Eqn.3.23), and the initial mesh (as well as

47
the underlying smooth limit surface) deforms continuously over time until an equilib¬
rium position is obtained. The deformation of the surface in response to the applied
forces is governed by the motion equation (Eqn.2.5). Within the physics-based mod¬
eling framework, the limit surface evolves as a consequence of the evolution of the
initial mesh. One can apply various types of forces on the limit surface to obtain
a desired effect, but the possible level of details appearing in a shape that can be
obtained through evolution is constrained by the number of control vertices in the
initial mesh. It might be necessary to increase the number of control vertices in the
initial mesh in order to obtain a detailed shape through this evolution process.
The number of control vertices defining the same smooth limit surface can be
increased by simply replacing the initial mesh with a mesh obtained after one subdi¬
vision step applied to the initial mesh. This new mesh has more number of vertices
but defines the same limit surface. Therefore, the dynamic Catmull-Clark surface
model can be subdivided globally to increase the number of vertices (control points)
of the model. For example, after one step of global subdivision, the initial degrees of
freedom p (refer to Eqn.3.15 and Eqn.3.16) in the dynamic system will be replaced
by a larger number of degrees of freedom q, where q = Ap. A is a global subdivi¬
sion matrix of size (M. N) whose entries are uniquely determined by Catmull-Clark
subdivision rules (see Section 3.1 for the details about the rules). Thus, p, expressed
as a function of q, can be written as
p = (ArAf Vq = Atq,
(3.24)

48
where At = (ArA)-1 Ar. Therefore, Eqn.3.15 and Eqn.3.16 can be rewritten as
s = (JAt)q, (3.25)
and
s(m, w,q) = (JAt)q. (3.26)
respectively. Xow the equation of motion needs to be derived for this new subdivided
model involving a larger number of control vertices namely q. The mass, damping
and stiffness matrices need to be recomputed for this “finer” level. The structure of
the motion equation as given by Eqn.2.5 remains unchanged, but the dimensionality
and the entries of M, D,K,p and fp change correspondingly in this newly obtained
subdivided level. In particular the motion equation, explicitly expressed as a function
of q, can be written as
M(?q + D(?q + K(?q = f9, (3.27)
where M, = / / fi.{A^)1 JrJAhiudv and the derivation of D^, and fq follow suit.
It may be noted that further subdivision, if necessary, can be carried out in a
similar fashion. Therefore, multilevel dynamic's is achieved through recursive subdi¬
vision on the initial set of control vertices. Users can interactively choose the level of
detail representation of the dynamic.1 model as appropriate for their modeling and de¬
sign requirements. Alternatively, the system can automatically determine the level of
subdivision most suitable for an application depending on some application-specific
criteria.

49
3.3 Finito Element Implementation
The evolution of the generalized coordinates for our new dynamic surface model
can be determined by the second-order differential equation as given by Eqn.2.5. An
analytical solution of the governing differential equation can not be derived in general.
However, an efficient numerical implementation can be obtained using the finite ele¬
ment method [42], The limit surface of the dynamic Catmull-Clark subdivision model
is a collection of bicubic patch elements. We use two types of finite elements for this
purpose - normal elements (bicubic patches assigned to the normal faces of the initial
mesh) and special elements (collection of infinite number of bicubic patches assigned
to each extraordinary vertex of the initial mesh). The shape functions for the nor¬
mal element are the basis functions of the bicubic surface patch, whereas the shape
functions for the special element are the collection of basis functions corresponding
to the bicubic patches in the special element. In the current implementation, the
M, D and K matrices for each individual normal and special elements are calculated
and they can be assembled into the global M, D and K matrices that appear in the
corresponding discrete equation of motion. In practice, we never assemble the global
matrices explicitly in the interest of time performance. The detailed implementation
is explained in the following sections.
3.3.1 Data Structures
The limit surface of the dynamic Catmull-Clark subdivision model is a collection
of bicubic patches, and hence requires us to keep track of the control polygons defining
such patches. We cían get the control polygons for the normal elements in the limit

50
Figuro 3.6. The data structure used for dynamic subdivision surface implementation.
surface from the initial control mesh itself. However, wo need to locally subdivide the
initial control mesh around the extraordinary vortices to obtain the control polygons
of the bicubic patches in the special elements on the limit surface.
A subdivision surface defined by a control mesh at any level is designed as a
class which has a pointer to its parent mesh, a sot of pointers to its offspring meshes
(arising out of local subdivision around the extraordinary vortices at that level), a list
of faces, edges, vortices and normal elements . Face, edge, vortex and normal elements
are, in turn, classes which store all the connectivity and other information needed to
either enumerate all the patches or locally subdivide around an extraordinary vortex
in that level. The implementation takes the initial mesh as the base subdivision
surface object (with its parent pointer sot to NULL) and locally subdivides the initial
mesh upto a user-defined maximum level around each extraordinary vortex to create

51
offspring objects at different levels (Fig.3.6). At this point, let’s take a closer look at
the normal and special element data structures and computation of the corresponding
local M, D and K matrices.
3.3.2 Normal Elements
Each normal element is a bicubic surface patch and hence is defined by 16
vertices (from the 8-eonneeted neighborhood of the corresponding normal face) in the
initial control mesh. Each normal element keeps a set of pointers to those vertices
of the initial mesh which act as control points for the given element. For a normal
element, the mass, damping and stiffness matrices are of size (16,16) and can be
computed exactly by carrying out the necessary integrations analytically. The matrix
J in Eqn.3.18, 3.20 and 3.22 needs to be replaced by 3P (of Eqn.3.2) for computation
of the local M, D and K matrices respectively of the corresponding normal element.
3.3.3 Special Elements
Each special element consists of an infinite number of bicubic patches in the
limit. We have already described a recursive enumeration of the bicubic patches of
a special element in Section 3.2.2. Let us now consider an arbitrary bicubic patch of
the special element in some level j. The mass matrix M., of this patch can be written
as
M.s = n'i'Mpns (3.28)
where Mp is the normal element mass matrix (scaled by a factor of A to take into
account of the area shrinkage in bicubic patches at higher level of subdivision) and

52
a is the transformation matrix of the control points of that arbitrary patch from the
corresponding control points in the initial mesh. The damping and stiffness matrices
for the given bicubic patch can be derived in a similar fashion. These mass, damping
and stiffness matrices from various levels of (local) subdivision can then be assembled
to form the mass, damping and stiffness matrices of the special element. It may be
noted that the stiffness energy due to plate terms diverges at the extraordinary points
on the limit surface. We solve the problem using a slightly different approach than
the one used in Halstead et al. [38]. When the area of the bicubic patch obtained via
local subdivision of the initial mesh around an extraordinary vertex becomes smaller
than the display resolution, the contribution from such a bicubic patch is ignored in
computing the physical matrices of the corresponding special element. The number
of extraordinary points in the limit surface is very few, and the above mentioned
approximation is found to work well in practice.
3.3.4 Force Application
The force f (u, v, t) in Eqn.3.23 represents the net effect of all externally applied
forces. The current implementation supports spring, inflation as well as image-based
forces. However, other types of forces like repulsion forces, gravitational forces etc.
can easily be implemented.
To apply spring forces, a spring of stiffness k can be connected from a point d0
to a point x0 on the limit surface (or, to the j-th level approximation mesh), the net

53
applied spring force being
f(x. t) = / A’(d0 — s(x. i))i(x — x0)dx.
(3.29)
where S is the unit impulse function implying f(x0, t) = At|do — s(x0, t) | and vanishes
elsewhere on the surface. However, the S function can be replaced with a smooth
kernel to spread the force over a greater portion on the surface. The spring forces can
be applied interactively using the computer mouse or the points from which forces
need to be applied can be read in from a file.
To recover shapes from 3D image data, we synthesize image-based forces. A 3D
edge detection is performed on a volume data set using the 3D Monga-Deriehe(MD)
operator [69] to produce a 3D potential field P(x,y,z), which we use as an external
potential for the model. The force distribution is then computed as
(3.30)
where A controls the strength of the force. The applied force on each vertex at the j-
th approximation level is computed by trilinear interpolation for evaluating Eqn.3.23
in Cartesian coordinates. More sophisticated image-based forces which incorporate
region-based information such as gradients of a thresholded fuzzy voxel classification
can also be used to yield better and more accurate shape recovery. It may be noted
that we can apply spring forces in addition with the image-based forces by placing

54
points on the boundary of the region of interest in each slices of the 3D volume (MR.
CT etc.) image data.
3.3.5 Discrete Dynamic Equation
The differential equation given by Eqn.2.5 is integrated through time by dis¬
cretizing the time derivative of p over time steps At. The state of the dynamic
subdivision surface at time t + Ad is integrated using prior states at time t and
t — Ai. An implicit time integration method is used in the current implementation
where discrete derivatives of p are calculated using
p(f + Ad) — 2p(f) + p(f — Ad)
Ad2
p(f + Ad)
(3.31)
and
(3.32)
The elemental mass, damping and stiffness matrices can be assembled to get the
global mass, damping and stiffness matrix for the smooth subdivision surface model.
However, we do not assemble these global sparse matrices explicitly for efficiency
reasons. For the time varying stiffness matrix, we recompute K at each time step.
Using Eqn.2.5, Eqn.3.31 and Eqn.3.32, the discrete equation of motion is obtained as
(2M + JAAd + 2Af2K)p(f + Ad) = 2Ad2ip(t + Ad) + (DAd — 2M)p(f — Ad) + 4Mp(f).
(3.33)

55
Figuro 3.7. Model subdivision to increase the degrees of freedom : (a) evolution of
the initial mesh, and (b) the corresponding limit surface evolution perceived by the
user.
This linear system of equations is solved iteratively between each time stop using the
conjugate gradient method [34, 75].
3.3.6 Model Subdivision
The initialized model grows dynamically according to the equation of motion
(Eqn.2.5). The degrees of freedom of the initialized model is equal to the number
of control vertices in the initial mesh as mentioned earlier. When an equilibrium is
achieved for the model, the number of control vertices can be increased by replacing
the original initial mesh by a new initial mesh obtained through one step of Catmull-
Clark subdivision. This increases the number of degrees of freedom to represent the
same (deformed) smooth limit surface and a new equilibrium position for the model
can be obtained. This process is depicted schematically in Fig.3.7. Model subdivision
might be needed to obtain a very localized effect on a smooth limit surface. For a

56
shape recovery application, one may start with a very simple initial model, and when
an approximate shape is recovered, the degrees of freedom can be increased to obtain
a new equilibrium position for the model with a better fit to the given data set. The
error of fit criteria for the discrete data is based on distancíe between the data points
and the points on the limit surface where the corresponding springs are attached.
In the context of image-based forces, if the model energy does not change between
successive iterations indicating an equilibrium for the given resolution, the degrees of
freedom for the model can be increased by the above-mentioned replacement scheme
until the model energy is sufficiently small and the change in model energy between
successive iterations becomes less than a pro-specified tolerance.
3.4 Generalization of the Approach
The? proposed approach cían be generalized for other approximating subdivision
schemes. However, a more general approach is presented in Chapter 5 for deriving the
dynamic framework. Dynamic Catmull-Clark subdivision surface model is reformu¬
lated in Section 5.2 using this general approach. A dynamic framework for another
popular approximating subdivision scheme namely, Loop’s subdivision scheme, is also
discussed in Chapter 5.

CHAPTER 4
DYNAMIC BUTTERFLY SUBDIVISION SURFACES
In the previous chapter, a dynamic.1 framework has been presented for an approx¬
imating subdivision scheme namely, Catmull-Clark subdivision scheme. In this chap¬
ter. a dynamic.1 framework is developed for (modified) butterfly subdivision scheme,
which is an interpolatory subdivision technique. First, a brief overview of the (mod¬
ified) butterfly subdivision scheme is presented. Next, a local parameterization tech¬
nique for the limit surface obtained via (modified) butterfly subdivision is discussed.
This parameterization scheme is then used to derive the dynamic.1 model. The imple¬
mentation details are also discussed.
4.1 Overview of the (Modified) Butterfly Subdivision Scheme
The butterfly subdivision scheme [30], like many other subdivision schemes used
in geometric design literature/applications, starts with an initial triangular mesh
which is also known as the control mesh. The vertices of the control mesh are known
as the control points. In each step of subdivision, the initial (control) mesh is refined
through the transformation of each triangular face into a patch with four smaller
triangular faces. After one step of refinement, the new mesh in the finer level retains
the vertices of each triangular face in the previous level and hence, interpolates the
coarser mesh in the previous level. In addition, every edge in each triangular face
57

58
Figuro 4.1. (a) The control polygon with triangular faces; (b) mesh obtained after
one subdivision stop using butterfly subdivision rules.
-W -W
(a) (b)
Figure 4.2. (a) The contributing vortices in the j-th level for the vortex in the j+1-
th level corresponding to the edge between v~[ and Vg; (b) the weighing factors for
different vortices.

59
is spilt by adding a now vertex whoso position is obtained by an affine combination
of the neighboring vortex positions in the coarser level. For instance, the mesh in
Fig.4.1(b) is obtained by subdividing the initial mesh shown in Fig.4.1(a) once. It
may bo noted that all the newly introduced vortices corresponding to the edges in the
original mesh have degree 6, whereas the position and degree of the original vortices
do not change in the subdivided mesh.
In the original butterfly scheme, the now vortices corresponding to the edges in
the previous level are obtained using an eight-point stencil as shown in Fig.4.2(a). To
show how this stencil is used, a vortex to bo introduced and the contributing vortices
from the stencil are highlighted in Fig.4.1(a). The name of the scheme originated
from the “butterfly”-like configuration of the contributing vortices. The weighing
factors for different contributing vortex positions are shown in Fig.4.2(b). The vortex
ejj1 in the j + 1-th level of subdivision, corresponding to the edge connecting vortices
v~[ and V2 at level j. is obtained by
ei2 1 = 0.5(v{ + v¿) + 2+ v|) - ?/;(v^ + v¿ + v¿ + v|), (4.1)
whore 0 < w < 1, and vj denotes the position of the ?'-th vortex at the j-th level.
The butterfly subdivision scheme producios a smooth C1 surface in the limit
except at the extraordinary points corresponding to the extra,ordinary vortices (vor¬
tices with degree not equal to 6) in the initial mesh [109]. All the vortices introduced
through subdivision have degree 6, and therefore, the number of extraordinary points
in the smooth limit surface equals the number of extraordinary vortices in the initial

60
w
(a) (b)
Figuro 4.3. (a) The weighing factors of contributing vortex positions for an edge
connecting two vortices of degree 6; (b) the corresponding case when one vortex is of
degree n and the other is of degree 6.

61
mesh. Recently, this scheme has been modified by Zorin et al. [109] to obtain better
smoothness properties at the extraordinary points. This modified scheme is known
as modified, butterfly subdivision technique. In Zorin et al. [109], all the edges have
been categorized into three classes : (i) edges connecting two vertices of degree 6 (a
10 point stencil, as shown in Fig.4.3(a), is used to obtain the new vertex positions
corresponding to these edges), (ii) edges connecting a vertex of degree 6 and a vertex
of degree n/6 (the corresponding stencil to obtain new vertex position is shown in
Fig.4.3(b), where q = .75 is the weight associated with the vertex of degree n ^ 6,
and Sj, = (0.25 + cos(2m/n) + Q.bcos(Am/n))/n, i = 0,1,..., n — 1, are the weights
associated with the vertices of degree 6), and (iii) edges connecting two vertices of
degree n ^ 6. The last case can not occur except in the initial mesh as the newly
introduced vertices are of degree 6, and the new vertex position in this last case is
obtained by averaging the positions obtained through the use of stencil shown in
Fig.4.3(b) at each of those two extraordinary vertices.
4.2 Formulation
In this section, a systematic.1 formulation of the dynamic.1 framework for the modi¬
fied butterfly subdivision scheme is presented. Unlike the approximating schemes, the
limit surface obtained via modified butterfly subdivision does not have any analytic.1
expression even in a regular setting. Therefore, derivation of a local parameterization
suitable for developing the dynamic.1 framework for the limit surface is the key issue
here which is discussed next.

62
Figuro 4.4. The smoothing effect of the subdivision process on the triangles of the
initial mesh.
4.2.1 Local Parameterization
The smooth limit surface defined by the modified butterfly subdivision technique
is of arbitrary topology whore a global parameterization is impossible. Nevertheless,
the limit surface can be locally parameterized over the domain defined by the initial
mesh following a similar approach described in Lounsbcry ot al. [53]. The idea
is to track any arbitrary point on the initial mesh across the meshes obtained via
the subdivision process (see Fig.4.4 and Fig.4.5). so that a correspondence can be
established between the point being tracked in the initial mesh and its imago on the
limit surface.
The modified butterfly subdivision scheme starts with an initial mesh consisting
of a set of triangular faces. The recursive application of the subdivision rules smoothes
out each triangular face, and in the limit, a smooth surface is obtained which can also
be considered as a collection of smooth triangular patches. The subdivision process
and the triangular decomposition of the limit surface is depicted in Fig.4.4. Note
that, the limit surface can be represented by the same number of smooth triangular
patches as that of the triangular faces in the initial mesh. Therefore, the limit surface

63
s can bo expressed as
n
s = 5>, (4.2)
k=1
whore n is the number of triangular faces in the initial mesh and is the smooth
triangular patch in the limit surface corresponding to the k-th triangular face in the
initial mesh.
The stage is now sot for describing the parameterization of the limit surface over
the initial mesh. The process is best explained through the following example. A
simple planar mesh shown in Fig.4.5(a) is chosen as the initial mesh. An arbitrary
point x inside the triangular face abc is tracked over the meshes obtained through
subdivision. The vortices in the initial mesh are darkly shaded in Fig.4.5. After
one stop of subdivision, the initial mesh is refined by addition of now vortices which
are lightly shaded. Another subdivision stop on this refined mesh loads to a finer
mesh with introduction of now vortices which are unshaded. It may bo noted that
any point inside the smooth triangular patch in the limit surface corresponding to the
fare abc in the initial mesh depends only on the vertices in the initial mesh which
are within the 2-neighborhood of the vertices a, b and c due to the local nature of
the subdivision process. For example, the vortex d, introduced after first subdivision
stop, can bo obtained using the 10 point stencil shown in Fig.4.3(a) on the edge ab.
All the contributing vortices in the initial mesh are within the 1-neighborhood of the
vortices a and b. A 10 point stencil can bo used again in the next subdivision stop
on the edge d.b to obtain the vortex g. Some of the contributing vortices at this level
of subdivision, for example, the (lightly shaded) 1-ncighbors of the vortex b (except

64
Figuro 4.5. Tracking a point x through various levels of subdivision : (a) initial
mesh, (b) the selected section (enclosed by dotted linos) of the mesh in (a), after one
subdivision stop, (c) the selected section of the mesh in (b). after another subdivision
stop.

65
d and e) in Fig.4.5(b), depend on some vertices in the initial mesh which are within
the 2-neighborhood of the vertices a, b and c in the initial mesh.
In the rest of the formulation, superscripts are used to indicíate the subdivision
level. For example, vjuvw denotes the collection of vertices at level j which control
the smooth patch in the limit surface corresponding to the triangular facie uvw at
the j-th level of subdivision. Let v°he be the collection of vertices in the initial mesh
which are within the 2-neighborhood of the vertices a, b and c (marked black in
Fig.4.5(a)). Let the number of such vertices be r. Them, the vector v°he, which is the
concatenation of the (x, y, z) positions for all the r vertices, is of dimension 3?’. These
r vertices control the smooth triangular patch in the limit surface corresponding to
the triangular face a,be in the initial mesh. Xow, there exists four subdivision matrices
(Aafcc),., (Aabc),, (Aabc)r and (Aa6c)m of dimension (3r, 3r) such that
"V adf
(A-ahc) fy aha
^bed
(Aabc)yabc,
Ví/e =
{-^-abc)r'vabc-
VÍe/ =
{A-abc)m'vabc;
(4.3)
where the subscripts t, /, r and m denote top, left, right and middle triangle positions
respectively (indicating the relative position of the new triangle with respect to the
original triangle), and v¿dy, v¿ed, v¿je and v¿ej are the concatenation of the (x.y.z)
positions for the vertices in the 2-neighborhood of the corresponding triangle in the

66
newly obtained subdivided mesh. The new vertices in this level of subdivision are
lightly shaded in Fig.4.5(b). The 2-neighborhood configuration of the vertices in the
newly obtained triangles is exactly the same as that of the original triangle, hence
local subdivision matrices are square and the vector dimensions on both sides of
Eqn.4.3 are the same. This concept is further illustrated in Fig.4.6.
Carrying out one more level of subdivision, a new set of vertices which are
unshaded in Fig.4.5(c) are obtained along with the old vertices. Adopting a similar
approach as in the derivation of Eqn.4.3, it can be shown that
Ydgi
(-A-bed)/ Y bp.d
<9 =
(A hed)yld
VL. =
( A-bfíd) r~V bed
V9 hi —
(A bed,) nibbed,
(4.4)
The relative position of the triangular face dgi in Fig.4.5(e) with respect to the
triangular face bed is topologically the same as of the triangular face adf in Fig.4.5(b)
with respect to the triangular face abc. Therefore, one can write (A^), = (A^),.
Using similar reasoning, Eqn.4.4 can be rewritten as
Vd§f = (A-bed) /.V'fcd = {A-abc) fYked,
VIhg = ( Afeed )/V¿erf = (A abc)Ybed
^eih (A-6ed)rVi)ed (-^-abc) bed,

67
(a)
(b)

68
(<0
(<1)
Figuro 4.6. Different sot of control points at a subdivided level obtained by applying
different subdivision matrices on a given sot of control points in a coarser mesh.

69
vsAi = (Abed)mv¡ed = (Aa6e)mv¿ed. (4.5)
Combining Eqn.4.3 and Eqn.4.5. it can bo shown that
( Aabc) /. ( Aabc); vabc:
iAabc) iiAabc) abe'
(Aabc)r(Aabc)i vfl,6e:
(Aabc)m(Aabc)iyabc (4-6)
Lot x bo a point with barycentric coordinates (o^hcJ^hcy^hc) inside the trian¬
gular face abc. When the initial mesh is subdivided, x becomes a point inside the
triangular face bed with barycentric coordinates (alPd, ftled, 7¿ed). Another level of
subdivision causes x to be included in the triangular face dgi with barycentric coor¬
dinates fjg/. 'g •)• Lot sJahc denote the j-th level approximation of the smooth
triangular patch sabc in the limit surface corresponding to the triangular face abc in
the initial mesh. Xow v°he can be written as
r r r
^ rjrt
"V abc. [dr: ^.r; dr: • • •: ®j/; fyi/; d/; • • •: bz, C: • • •]
where the subscripts x, y and 2 indicíate the x, y and z coordinates respectively of the
corresponding vertex position. The expressions for v¿ed and v¿gi can also be written
dgi
’ bhg
'Vfíih
ghi

70
in a similar manner. Next, the matrix B°he can be constructed as follows:
r r
r r
The matrices Bled and B^ can also be constructed in a similar fashion. Xow s°he(x),
sa6c(x): an(l sa6c(x) can 4)0 written as
SlbAx)
= BL(X)VL>
Slbc(x)
= BL(X)Vld
= BLd(X)(A«te)iVL>
SlhAx)
= BdSi(x)Vrfsi
= Bd9f(x)(Aa6e),V¿ed = B^íxJÍAateJ^A^)^
(4.7)
Proceeding in a similar way, the expression for j-th level approximation
of snfoc(x), is given by
j
Sabc(x) = Bwm;(x) (Aabc)m ■ ■ ■ (Aabc)¿(Aabc)ivabc
= Bívw(x)(Aíbc)Vabc
= BtbrÁX)Vabc> (4‘8)
where x is inside the triangular face uvw at level j (with an assumption that uvw is the
triangular face in the middle with respect to its coarser level original triangular face in

71
the previous level), (A^) = {Aahc)m ... (Aabc),(Aabc)l and B¿6e(x) = B{0l0(x)(A^).
It may be noted that the sequence of applying (A^),. (AafIC)l, (Aafjc)r and {Aabc)m
depends on the triangle inside which the tracked point x falls after each subdivision
step. Finally, the local parameterization process can be completed by writing
S«,6e(x) = ( lim B^e(x))v°6e = Bafcfxjv^. (4.9)
j~>0O
In the above equation, is the collection of basis functions at the vertices of
v°he. It may also be noted that the modified butterfly subdivision scheme is a sta¬
tionary subdivision process, and hence new vertex positions are obtained by affine
combinations of nearby vertices. This guarantees that each row of the matrices
(Aabc)p (Aabc)i, (Aa/,c)r and {Aabc)m sums to one. The largest eigenvalue of such ma¬
trices is 1 and therefore the limit in Eqn.4.9 exists. Xow, assuming the triangular
face abc is the Avth face in the initial mesh, Eqn.4.9 can be rewritten as
s*(x) = B/fe(x)v2 = B/feixJA/fep, (4.10)
where p is the concatenation of the (x,y,z) positions of all the vertices in the initial
mesh and the matrix A*, when post-multiplied by p, selects the vertices v° controlling
the k-th smooth triangular patch in the limit surface. If there are t vertices in the
initial mesh and r of them control the A’-th patch, then p is a vector of dimension 31,
Ak is a matrix of dimension (3?’, 31), and B*(x) is a matrix of dimension (3, 3?’).

72
Combining Eqn.4.2 and Eqn.4.10. it can bo shown that
n
s(x) = B^(x)A^)p = J(x)p. (4.11)
k=1
where J, a matrix of dimension (3,31), is the collection of basis functions for the
corresponding vortices in the initial mesh. The vector p is also known as the degrees
of freedom vector of the smooth limit surface s.
4.2.2 Dynamics
An expression of the limit surface obtained via modified butterfly subdivision
process is derived in the last section. Xow a dynamic framework for the limit surface
can bo derived using this expression. The derivation of the dynamic.1 model from this
point is exactly similar in spirit as that of the dynamic.1 Catmull-Clark subdivision
model presented in Section 3.2.4. The vortex positions in the initial mesh defining
the smooth limit surface s are treated as a function of time in order to embed the
modified butterfly subdivision model in a dynamic.1 framework. The velocity of this
surface model can bo expressed as
s(x, p) = J(x)p, (4.12)
whore an ovcrstruck dot denotes a time derivative and x e S°, 5*° being the domain
defined by the initial mesh. As in the case of the dynamic.1 Catmull-Clark subdivision
surfaces, the Lagrangian equation of motion (Eqn.2.1) is used to derive the dynamic
modified butterfly subdivision surface model.

73
Lot /i bo the mass density function of the surface. Then the kinetic energy of
the surface is
T = ^ f /.¿(xjs2 (x)s(x) whore (using Eqn.4.12) M = fxeSn /.i(x)Jrj (x)J(x) sion (3t,3t). Similarly, let 7 bo the damping density function of the surface. The
dissipation energy is
F = \ [ 7'(x)sri (x)s{x)dx = ^p^Dp, (4.14)
2 Jxesn 2
whore D = /xe5n y(x)Ji (x)J(x) potential energy of the smooth limit surface can be expressed as
U = ^prKp, (4.15)
whore K is the stiffness matrix of dimension (31 31) obtained by assigning various
internal energies to a discretized approximation of the limit surface and is detailed
in Section 4.3.
Using the expressions for the kinetic, dissipation and potential energy in Eqn.2.1,
the same motion equation as in Eqn.2.5 can be obtained. The generalized force vector

74
fp can bo obtained is a similar fashion described in Section 3.2.4 and is given by
(4.16)
4.2.3 Multilevel Dynamics
The initial mesh of the dynamic modified butterfly subdivision surface model
can bo subdivided to increase the degrees of freedom for model representation. The
situation is exactly similar to that of the dynamic Catmull-Clark subdivision surface
model as described in Section 3.2.5 and a similar example is used for illustration.
After one stop of modified butterfly subdivision, the initial degrees of freedom p (refer
to Eqn.4.11 and Eqn.4.12) in the dynamic system is replaced by a larger number of
degrees of freedom q, whore q = Bp. B is a global subdivision matrix of size (3s. 31)
whoso entries are uniquely determined by the weights used in the modified butterfly
subdivision scheme (see Section 4.1 for the weights). Thus, p, expressed as a function
of q, can bo written as
p = (BrB) 1Biq = Btq,
(4.17)
whore B^ = (BrB) B7. Therefore, Eqn.4.11 and Eqn.4.12 is modified as
s(x) = (J(x)Bf)q,
(4.18)
and
s(x,q) = (J(x)Bf)q
(4.19)

75
Figuro 4.7. (a) An initial mesh, and (b) the corresponding limit surface. The domains
of the shaded elements in the limit surface are the corresponding triangular faces in
the initial mesh. The encircled vertices in (a) are the degrees of freedom for the
corresponding element.
respectively. The motion equation, explicitly expressed as a function of q, can be
written as
M(?q + D(?q + K(?q = f,, (4.20)
whore M9 = fxe$i /i(x)(B^f J1 (x)J(x)BVix, 51 being the domain defined by the
newly obtained subdivided mesh. The derivations of D^, and are similar.

76
4.3 Finito Element Implementation
In the previous section, the smooth limit surface obtained via modified butterfly
subdivision process is expressed as a function of the control vertex positions in its
initial mesh; mass and damping distribution, internal deformation energy and forces
are assigned to the limit surface in order to develop the corresponding physical model.
In this section, the implementation of this physical model is detailed. It may be noted
that even though the formulation of the dynamic framework for modified butterfly
subdivision scheme is similar to that of Catmull-Clark subdivision scheme once the
local parameterization is derived, the finite element implementation is totally different
as the limit surface does not have any analytic expression in case of the modified
butterfly subdivision scheme.
In Section 4.2 it was pointed out that the smooth limit surface obtained by the
recursive application of the modified butterfly subdivision rules can be represented by
a set of smooth triangular patches, each of which is represented by a finite element.
The shape (basis) function of this finite element is obtained by smoothing a hat
function through repeated application of the modified butterfly subdivision rules.
The number of finite elements in the smooth limit surface is equal to the number
of triangular faces in the initial mesh as mentioned earlier (refer Fig.4.4 and 4.7).
A detailed discussion on how to derive the mass, damping and stiffness matrices
for these elements is presented next. These elemental matrices can be assembled
to obtain the global physical matrices M, D and K, and a numerical solution to
the governing second-order differential equation as given by Eqn.2.5 can be obtained

77
using finito element analysis techniques [42], The same example as in Section 4.2
(refer Fig.4.5) is used to develop the related concepts. The concept of elements along
with the control vertices and their corresponding domains is further illustrated in
Fig.4.7.
Xow it will bo shown how to derive the mass, damping and stiffness matrices
for the element corresponding to the triangular face abc in Fig.4.5. The derivations
hold for any element in general.
4.3.1 Elemental Mass and Damping matrices
The mass matrix for the element given by sat¡c (Eqn.4.9) can be written as
Mabc= f /¿(xHB^x^jB^x^fix. (4.21)
JxEsnhc
However, from Eqn.4.9 it is known that the basis functions corresponding to the
vertices in v°he which are stored as entries in ~Bahc are obtained as a limiting process.
Those basis functions do not have any analytic.1 form, hence computing the inner
product of such basis functions as needed in Eqn.4.21 is a challenging problem. In
Lounsbcry ct al. [53], an outline is provided on the computation of those inner
products without performing any integration. In this dissertation, a different yet
simpler approach is developed to solve this problem. The smooth triangular patch in
the limit surface corresponding to the face abc in the initial mesh is approximated by
a triangular mesh with 4J faces obtained after j levels of subdivision of the original
triangular face a,be (each subdivision stop splits one triangular face into 4 triangular

78
faces). Then the mass matrix can be expressed as
Mafic
E
/í(x){B¿í)e(x)}i{B¿í)e(x)}dx.
(4.22)
The j-th level approximation of the corresponding basis functions can be explicitly
evaluated (refer Eqn.4.8 for an expression of B¿he). An important point to note is
that Eqn.4.8 involves several matrix multiplications and hence can be very expensive
to evaluate. However, the matrix (A3abc)(= (Aabc)m ... (A^J^Aai,.),) in Eqn.4.8
encodes how vertices in the 2-neighborhood of the triangular face uvw at level j are
related to the vertices in the 2-neighborhood of the triangular face abc in the initial
mesh. In the implementation, how a new vertex is obtained from the contributing
vertices in its immediate predecessor level is tracked. The information stored in (A3abc)
can be obtained by tracking the contributions from level j to level 0 recursively. Thus
the entries of (AJabc) can be determined without forming any local subdivision matrices
and thereby avoiding subsequent matrix multiplications.
By choosing a sufficiently high value of j. a reasonably good approximation of
the elemental mass matrices is achieved. The computations involved in evaluating
the integrals in Eqn.4.22 are eliminated by using discrete mass density function which
has non-zero values only at the vertex positions of the j-th subdivision level mesh.
Therefore, the approximation of the mass matrix for the element can be written as
= E //(V'/H B'^Av'/)}' {B/if)r {v/)}
i=l
(4.23)

79
where k is the number of vertices in the triangular mesh with 4J faces. This approxi¬
mation has been found to be very effective and efficient for implementation purposes.
The elemental damping matrix can be obtained in a similar fashion.
4.3.2 Elemental Stiffness Matrix
The internal energy is assigned to each element in the limit surface, thereby
defining the internal energy of the smooth subdivision surface model. A similar
approach as in the derivation of the elemental mass and damping matrix is taken
and the internal energy is assigned to a j-th level approximation of the element.
Three types of internal energy are used - tension, stiffness and spring energy.
For the examples used throughout the paper, this energy at the j-th level of approx¬
imation can be written as
Eabc ~ EJabc — (EJahc)t + (EJahf,)H + (T’a6e).,p
(4.24)
where the subscripts t, st and sp denote tension, stiffness and spring respectively.
The expression for the tension energy, which is essentially equivalent to the first
order strain (membrane) energy [96], is
Q
1
(4.25)
where kt. is a constant, v/ and vj^, the /-th and m-th vertex in the j-th level mesh,
are in the 1-neighborhood of each other, Q is the domain defined by all such vertex

80
pairs, and vJabc is the concatenation of the (x.y.z) positions of all the vertices in the
j-th subdivision level of the triangular face abc in the initial mesh.
Similarly, the expression for stiffness energy, which is equivalent to the second
order strain (thin plate) energy [96], can be written as
(Eibc)s, = ^wElv/^2vm+vn|2
z n
= (Kibc) sMbc}: (4‘26)
where vf, v^ and v£ are the three vertices of a triangular face. The summation in¬
volves three terms corresponding to each triangular face, and is over all the triangular
faces in the mesh at the j-th level of subdivision.
Finally, a spring energy term is added which is given by
sp
(hm)sp(
I v? — V*?
1 v l m
| f/m)
|
i i 1
v/ - vJm\
-E
z n
K
(4.27)
where (kim) is the spring constant, Q is the domain as in Eqn.4.25 and ('im is the
natural length of the spring connected between v/ and v^. It may be noted that
the entries in (K.3abc) depend on the distancíe between the connected vertices and
hence, (K¿6<.) , unlike other elemental matrices, is a function of time which needs to
be recomputed in each time step.

81
Combining the expressions for tension, stiffness and spring energy, it can be
shown that
EL = ¿Kfc/WKL), + MKÍ + (Ki,Js„}M J
= ¿MíTfKiJKiJ
= ^vll'híAÍ^j'hKivXAi^Jílvl}, (4.28)
where (AJabc) and v°he are same as in Eqn.4.8. Therefore, the expression for the
elemental stiffness matrix is given by
Ka„ = (A{hf{K{lx)(A{h,:). (4.29)
It may be noted that the matrix multiplications for constructing K.abc are avoided in
the implementation by following the same technique described in Section 4.3.1.
4.3.3 Other Implementation Issues
The techniques of applying synthesized forces on the smooth limit surface ob¬
tained via modified butterfly subdivision process are similar to those described in
Section 3.3.4 in the context of dynamic Catmull-Clark subdivision surface model.
The techniques of model subdivision and discrete dynamic equation derivation are
also the same.

82
4.4 Generalization of the Approach
The proposed approach of deriving the dynamic.1 framework for modified butter¬
fly subdivision scheme is generalized for other interpolator)- subdivision schemes in
the next chapter. Chapter 5.

CHAPTER 5
UNIFIED DYNAMIC FRAMEWORK FOR SUBDIVISION SURFACES
Dynamic.1 framework for specific.' subdivision schemes was presented in the last
two chapters. In Chapter 3, Catmull-Clark subdivision scheme, a representative of
the approximating subdivision schemes, is embedded in a physics-based modeling en¬
vironment. In Chapter 4, a dynamic.' framework is provided for the modified butterfly
subdivision scheme, a popular interpolatory subdivision technique. In this chapter,
a unified approach is presented for embedding any subdivision scheme in a dynamic.'
framework.
5.1 Overview of the Unified Approach
The key concept of the unified approach is the ability to express the smooth
limit surface generated by a subdivision scheme as a collection of a single type of
finite elements. The type of this finite element depends on the subdivision scheme
involved. For example, the limit surface generated by the Catmull-Clark subdivision
scheme can be expressed as a collection a quadrilateral finite element patches. This
approach differs form the one taken in Chapter 3 where the limit surface was a
collection of normal elements (corresponding to the quadrilateral bicubic B-spline
patches away from the extraordinary points) and special elements (corresponding to
83

84
the n-sided patches near extraordinary vertices of degree n). This concept will be
further elaborated in the next few sections.
The unified approach adopts two different strategies depending on the subdivi¬
sion scheme involved. One strategy is for the approximating schemes, whose limit
surface is some type of B-spline surface away from the extraordinary points, and
the other is for the interpolator)- subdivision schemes, where the limit surface does
not have any analytical expression. It may be recalled that recursive subdivision is
a method for smoothing polyhedra, and hence corresponding to each n-sided face
in the control mesh, there would be a smooth n-sided patch in the limit surface
in general. For example, the control mesh (after at most one subdivision step) of
the Catmull-Clark subdivision scheme consists of quadrilateral faces, which when
smoothed through subdivision lead to quadrilateral patches in the limit surface (see
Fig.5.1(b)). In the ease of approximating subdivision schemes, the faces which do
not contain any extraordinary vertex lead to spline patches with analytic.1 expressions.
On the other hand, faces which have one extraordinary vertex, lead to patches whose
analytic.1 expressions are difficult to determine. Continuing the example with the
Catmull-Clark subdivision scheme, the faces without extraordinary vertices lead to
quadrilateral bicubic B-spline patches in the limit, whereas the faces with one extraor¬
dinary vertex lead to quadrilateral patches whose analytic.1 expressions are difficult
to obtain. In Chapter 3, n such adjacent patches corresponding to an extraordinary
vertex of degree n are effectively combined together to obtain an expression for the re¬
sulting n-sidod patch (see Fig.5.1(a)). Recently, Stain [90, 91] has introduced schemes

85
for exactly evaluating the limit surface, even near the extraordinary points, for ap¬
proximating subdivision schemes. Analytic expressions for the patches resulting from
smoothing faces with an extraordinary vertex can be obtained using these schemes.
Therefore, the limit surface can be expressed as a collection of a single patch type,
and consequently a single type of finite element can be used to provide the dynamic
framework for the approximating subdivision schemes. This unified framework for
approximating subdivision schemes is fully worked out in this chapter for Catmull-
Clark and Loop subdivision schemes and a general outline is also presented on how
to carry out the same strategy for other approximating subdivision schemes.
The limit surface resulting from an interpolatory subdivision scheme does not
have an analytic expression in general, and hence a strategy similar to the one adopted
in Chapter 4 needs to be used. An outline of the strategy to be used for interpolatory
subdivision schemes is also presented in this chapter.
5.2 Unified Dynamic Framework for Catmull-Clark Subdivision Surfaces
A systematic formulation along with the implementation details of the unified
dynamic framework for Catmull-Clark subdivision surfaces is presented in this sec¬
tion. The key difference between the framework developed in Chapter 3 and the one
presented here is the representation of the limit surface, which leads to different types
of finite elements used for developing the dynamic.' framework. This is illustrated with
a schematic diagram in Fig.5.1.

86
(a) (b)
Figuro 5.1. A control mesh with an extraordinary vortex of degree 5 and the corre¬
sponding limit surface : (a) using the concepts developed in Chapter 3, whore the
limit surface consists of quadrilateral normal elements and a pentagonal special ele¬
ment; (b) using the unified concept developed in this chapter, whore the limit surface
consists of one single typo of quadrilateral finite element.

87
Figuro 5.2. In Catmull-Clark subdivision, each non-boundary quadrilateral face in
the control mesh has a corresponding quadrilateral patch in the limit surface : (a)
control mesh, (b) limit surface.
Following the concepts developed in Chapter 3, the limit surface of the control
mesh shown in Fig.5.1. consists of quadrilateral bicubic B-splinc patches correspond¬
ing to the faces marked hi’ (faces with no extraordinary points), and a pentagonal
patch corresponding to the faces marked ’s’ (faces having one extraordinary vertex
of degree 5) (Fig.5.1(a)). However, in this section, it will be shown that the entire
limit surface can be expressed as a collection of quadrilateral patches as illustrated in
Fig.5.1(b) using the schemes proposed by Stam [90]. Next, the local parameterization
of the limit surface is described in order to develop the unified dynamic.' framework.

5.2.1 Local Parameterization
As mentioned earlier, the control mesh (after at most one subdivision step) for
the Catmull-Clark subdivision scheme consists of quadrilateral faces which lead to
quadrilateral patches in the limit surface. For the sake of simplicity, it is assumed
that each face has at most one extraordinary vertex. If this assumption is violated,
then one more subdivision step needs to be performed on the current control mesh
in order to obtain a new control mesh on which the following analysis can be carried
out. The number of quadrilateral patches in the limit surface is equal to the number
of non-boundary quadrilateral faces in the control mesh (Fig.5.2). Therefore, the
smooth limit surface s can be expressed as
n
s = 5>, (5.1)
1=1
where n is the number of non-boundary faces in the control mesh and S; is the smooth
quadrilateral patch corresponding to the /-th non-boundary quadrilateral face in the
control mesh. Each of these quadrilateral patches can be parameterized over the
corresponding non-boundary quadrilateral face in the control mesh. However, since a
quadrilateral face can easily be parameterized over a [0, l]2 domain, each quadrilateral
patch is locally parameterized over the domain [0, l]2.
The non-boundary quadrilateral faces are of two types : (a) faces having no
extraordinary vertices (dubbed as “regular” faces in Section 3.2.1, marked as k in
Fig.5.2(a)) and (b) faces with one extraordinary vertex (dubbed as “irregular” faces
in Section 3.2.2, marked as ( in Fig.5.2(a)). If there are m regular and n — m irregular

faces, then Eqn.5.1 can be rewritten as
rn n—m
s = Es?+E^ (5-2)
¿=i j=i
where s, is the quadrilateral patch corresponding to the ?!-th regular face and sj is
the quadrilateral patch corresponding to the j-th irregular face.
The quadrilateral patch in the limit surface corresponding to each regular face
is a bicubic B-spline patch, which is defined over [0, l]2. The set of control vertices
defining this bicubic B-spline patch can be obtained using the adjacent face informa¬
tion (refer to Section 3.2.1 and Fig.3.1 in Chapter 3). Therefore, the quadrilateral
patches in the smooth limit surface corresponding to the regular faces in the con¬
trol mesh can be easily expressed analytically, which are essentially bicubic B-spline
patches defined by 16 control vertices over a [0, l]2 domain. The analytic.1 expression
for the quadrilateral patch corresponding to the regular face i is given by
S'f = Jfc(M5w)Pi
= (Jft('M, e)Aj)p
= 3i(n. r)p. (5.3)
where 0 < u. v < 1, Jh(u,v) is the collection of the bicubic B-spline basis functions,
p,¿ is the concatenation of the 16 control vertex positions defining the bicubic B-spline
patch, A,¿ is a selection matrix which when multiplied with p, the concatenation of

90
all the control vortex positions defining the smooth limit surface, selects the corre¬
sponding sot of control vortices, and J¿(m, v) = J<,(■«, ”>A„
The analytic.1 expression of the quadrilateral patches corresponding to the ir¬
regular faces in the control mesh was difficult to derive, and hence an alternative
approach was taken in Mandal ot al. [57, 60] and Qin ot al. [76] as well as in Chapter
3. However, very recently an efficient scheme for evaluating Catmull-Clark subdivi¬
sion surfaces at arbitrary parameter values was proposed by Stam [90]. The proposed
approac.ii, involving eigen-analysis of the subdivision matrix, loads to an analytic.' ex¬
pression of a quadrilateral patch parameterized over an irregular face. Following the
scheme developed by Stam [90], the quadrilateral patch corresponding to the irregular
face j is given by
s, = •!•'dP,
= (Jd*(«5w)A/)P
= Jj(u,v)p, (5.4)
whore 0 < u. v < 1 as before. Jdk(u,v) is the collection of basis functions for the
corresponding quadrilateral patch in the smooth limit surface. The subscript <4 is
used to denote the fact that the irregular face has an extraordinary vortex of degree
k. The detailed derivation and the analytic.1 expressions of those basis functions
involving the eigenvalues and eigenvectors of the subdivision matrix can bo found in
Stam [90]. The rest of the notations used in Eqn.5.4 have the usual moaning : pj is
the concatenation of the 2A: + 8 control vertices defining the quadrilateral patch in the

91
Figuro 5.3. (a) The marked 16 control vortices define the shaded quadrilateral patch
associated with the shaded regular face in the control mesh; (b) the marked 14 control
vortices define the shaded quadrilateral patch associated with the shaded irregular
face in the control mesh.
limit surface, p is the concatenation of all the control vortex positions defining the
smooth limit surface. Aj is a selection matrix which when multiplied with p selects
the corresponding sot of control vortices, and Jj(«, v) = J4 («, v)Aj.
It may bo noted that the number of control vortices in the initial mesh defining
a quadrilateral patch in the smooth limit surface is 2k + 8, whore k = 4 in case
the associated quadrilateral face in the control mesh is regular, or k = degree of the
extraordinary vortex if the associated quadrilateral face is irregular. For example, the
shaded quadrilateral patch is associated with the shaded regular face in Fig.5.3(a),
and the 16 control vortices defining this patch (which is actually a bicubic B-splinc
patch) are marked. Similarly, the shaded quadrilateral patch is associated with the

92
shaded irregular face in Fig.5.3(b), and the 14 control vertices defining this patch are
highlighted. Now an expression of the smooth limit surface can be easily formulated.
Using Eqn.5.2, 5.3 and 5.4, it can be shown that
rn n—m
s = 5Zjíp+5Zjjp
¿=i j=i
m n—m
= (Ejí+ E Ji)p
¿=i j=i
= Jp, (5.5)
where J = i J; + Z)"=r J?)- ^ote that even though the initial mesh serves as the
parametric domain of the smooth limit surface, each quadrilateral face in the initial
mesh and consequently the smooth limit surface can be defined over a [0, l]2 domain.
5.2.2 Dynamics
The expression of the limit surface as given by Eqn.5.5 is same as the one derived
in Chapter 3 (Eqn.3.15). However, the contents of the matrix J is totally different in
two cases. Nevertheless, the dynamic's of the limit surface can be formulated using
the same symbolic.1 expressions (which, when actually evaluated will lead to different
expressions depending on the contents of the matrix J). Therefore, the symbolic.1
expression of the motion equation remains the same as in the earlier formulation of
the dynamic.1 Catmull-Clark subdivision surface model.
5.2.3 Finite Element Implementation
In the unified dynamic.1 framework, the smooth limit surface obtained via Catmull-
Clark subdivision process is expressed as a collection of quadrilateral patches. Each

93
quadrilateral patch in the limit surface is considered as a finite element. Therefore,
the limit surface under the unified framework consists of a single type of finite el¬
ements instead of two different types of elements as in Mandal ot al. [57, 60] and
Qin ot al. [76]. The mass, damping and stiffness matrices for those elements along
with the generalized force vector can bo evaluated analytically, provided the mate¬
rial properties (like mass, damping, rigidity and bonding distributions) have analytic.1
expressions. Those distribution functions are assumed to bo constant in most (.'ases.
Xow, the expressions of the mass, damping and stiffness matrices for a quadrilateral
element which is a bicubic B-splinc can bo written as
and
(5.6)
D,
1 rl
0 JO
qjl Jtidudv,
(5.7)
Ke
[ [ («lliíJh)'»}1 {(J/¡)a} + «'22 { ( ),, } 7 { (J ft),,, } + /^l 1 { (J 6) MM } 7 { ( J 6)« « }
Jo Jo
+A2{(J/>)m-<.}:/ {(J ft)«} + /^{(Jfi)?™}2 {(J b)w})dudv (5-8)
respectively, whore Jh is the bicubic B-splinc basis matrix, /./,(«, v) is the mass density,
q(«, v) is the damping density, a ¡¡{a. r) and ¡¡¡{o. r) are the tension and rigidity
functions respectively. The subscript a and v denote partial derivatives with respect
to u and v respectively. The subscript e is used to indicíate elemental matrices which

94
aro of size (16,16). The expression for the corresponding generalized force vector is
(5.9)
The mass, damping and stiffness matrices for the quadrilateral elements which
are not bicubic B-splincs (corresponding to the irregular faces) can bo expressed
analytically as well simply by replacing the matrix J<, in Eqn.5.6, 5.7 and 5.8 by
the matrix J^ (refer to Eqn.5.4), whore k denotes the degree of the extraordinary
vortex associated with the corresponding irregular face. Those elemental matrices
are of size (2k + 8, 2k + 8). The generalized force vector for those elements can also
bo determined in a similar fashion. It may bo noted that the limits of integration
need to bo chosen carefully for elemental stiffness matrices as the second derivative
diverges near the extraordinary points for Catmull-Clark subdivision surfaces. The
elemental matrices combined together, either explicitly or implicitly, give the global
matrices. Similarly, elemental generalized force vectors can bo combined together to
obtain the global generalized force vector. The motion equation is discretized and
solved following similar techniques described in Chapter 3.
It may bo noted that oven though an analytical expression for a quadrilateral
element in the limit surface which is not a bicubic B-splinc exists, it is cumbersome
to actually evaluate the elemental matrix expressions. Numerical integration using
Gaussian quadrature may bo used to obtain approximations of those elemental ma¬
trices. However, in this dissertation, an approach similar to the dynamic modified

95
butterfly subdivision model presented in Chapter 4 is taken for implementation pur¬
poses. An approximation of the smooth limit surface is obtained by subdividing
the initial control mesh j times, and a spring-mass system is developed on this j-th
approximation level in a similar fashion as in Section 4.3. The physical matrices
of this system are used as approximations to the actual physical matrices. These
approximations have been found to be very efficient for implementation purposes.
5.3 Unified Dynamic Framework for Loop Subdivision Surfaces
Loop’s subdivision scheme starts with a triangular control mesh and generates a
smooth surface with triangular patches in the limit. It is an approximating subdivi¬
sion scheme which generalizes recursive subdivision schemes for obtaining C2 quartic
triangular B-spline patches in a regular setting. In each step of Loop subdivision, each
(non-boundary) triangular face is refined into 4 triangular faces using the following
rules :
• For each (non-boundary) vertex V of degree n, a new vertex point is introduced.
The position of this newly introduced vertex point is given by Q:(")v+vi+-+v";
where v is the position vector of vertex V, Vi,..., vn are the vertex positions
of the n vertices connected to vertex V. a(n) = and ft(n) = | —
(?>+2cns(2n / n))2
64 '
• For each (non-boundary) edge E, a new edge point is introduced. Let E be the
connecting edge between vertices \\ and V2, and is shared by faces ip and F2.
If Fi and F2 have vertices V'iy and V>2 respectively (apart from \\ and V2), then
the position of the newly introduced edge point is given by 3(Vi+V2)+Vf1+vf2 _

96
Figuro 5.4. (a) The control polygon with triangular faces; (b) mesh obtained after
one subdivision stop using Loop’s subdivision rules.
whore vl5 v2. vFl and vFa are the position vector of the vortex \\, V'2, VFl and
Vt\¿ respectively.
• Xow edges are formed by connecting each now vortex point to the now edge
points corresponding to the edges incident on the old vortex, and by connecting
each now edge point to the now edge points of the other edges in the two faces
which shared the original edge.
• Xow faces are defined as faces enclosed by the now edges.
Examples of refining an initial mesh using Loop’s subdivision rules are shown
in Fig.5.4 and 5.5. Those subdivision rules ensure tangent piano continuity of the
limit surface oven in a irregular sotting, i.o., when the triangular control mesh has

97
extraordinary vertices whose degree is not equal to 6. The limit surface shown in
Fig.2.1(d) is actually obtained by subdividing the control mesh shown in Fig.2.1(a)
using Loop subdivision rules. A detailed discussion on how to obtain positions and
normals in the smooth limit surface generated by the Loop subdivision scheme can
be found in Section 4.2 of Hoppe’s thesis [39].
5.3.1 Local Parameterization
The limit surface obtained via Loop’s subdivision scheme can be locally pa¬
rameterized easily. This local parameterization scheme is very similar in nature to
the one described for Catmull-Clark subdivision scheme in Section 5.2.1. For Loop’s
scheme, the smooth limit surface consists of triangular patches and the number of
these triangular patches is same as the number of non-boundary triangular faces in
the control mesh. Therefore, each of the triangular patch in the limit can be locally
parameterized over the corresponding triangular face in the control mesh. In may
be noted that each triangular face in the control mesh can be parameterized over a
triangular domain whose vertices are located at (0,0), (0,1) and (1,0), and hence
each triangular patch and consequently the smooth limit surface can be defined over
this domain (refer Fig.5.6).
The triangular patches in the smooth limit surface are of two types. For a non¬
boundary triangular face in the control mesh with no extraordinary vertices (i.e., with
three vertices of degree 6), the corresponding triangular patch in the limit surface is
a particular type of triangular B-spline (the three-direction quartic box spline) whose
analytic.1 expression is easy to obtain. This triangular B-spline patch is controlled by

98
Figuro 5.5. An initial mesh and the corresponding limit surface obtained using Loop’s
subdivision rules. The domains of the shaded triangular patches in the limit surface
are the corresponding triangular faces in the initial mesh. The encircled vertices are
the control vertices for the corresponding triangular patch in the limit surface.

99
Figuro 5.6. Each triangular patch in the limit surface can bo associated with a non¬
boundary triangular face in the initial mesh, which in turn can bo parameterized over
a triangle with vortices at (0, 0), (1.0) and (0,1).

100
12 vortices as shown in Fig.5.5 (the set of enclosed vertices in the left hand side). The
triangular patch in the limit surface corresponding to a non-boundary triangular face
in the control mesh with one extraordinary vertex can also be expressed analytically
using the schemes proposed by Stain [91]. This triangular patch is controlled by n + 6
vertices in the control mesh where n is the degree of the extraordinary vertex. The
set of control vertices for a triangular patch of the later type is shown in the right
hand side of Fig.5.5. Therefore, each triangular patch in the limit surface can be
expressed analytically, and an expression of the limit surface similar to Eqn.5.5 can
be obtained.
5.3.2 Dynamics
Once an expression for the limit surface obtained via Loop’s subdivision is ob¬
tained, the dynamic model can be developed following an exactly similar procedure
described for Catmull-Clark subdivision scheme in Section 5.2.2. The motion equa¬
tion of the dynamic Loop subdivision model can also be derived in a similar fashion.
5.3.3 Finite Element Implementation
The implementation of the dynamic framework for Loop subdivision scheme us¬
ing the unified approach treats each triangular patch in the limit surface as a finite
element. Each triangular patch has an analytic expression, and hence the elemental
physical matrices and the generalized force vector can be derived analytically. The
derivation of an exact expression for elemental matrices is cumbersome for the tri¬
angular patches corresponding to the triangular faces with an extraordinary vertex,
and numerical integration using Gaussian quadrature may be used for deriving an

101
approximation. However, a practical alternative for implementation is to subdivide
the control mesh j times using Loop’s subdivision rules, and to build a spring-mass
system on this j-th level approximation as has been done for the dynamic modified
butterfly subdivision model in Section 4.3. The physical matrices of this spring-mass
system provide an approximation of the original physical matrices, and it works well
in practice.
5.4 A General Outline of the Framework for Approximating Subdivision Schemes
The unified approach presented to provide a dynamic framework for Catmull-
Clark and Loop subdivision schemes can be generalized easily. This approach involves
three steps which are as follows :
1. The limit surface obtained via an approximating subdivision scheme can be
expressed as a collection of smooth patches which can be locally parameterized
over a corresponding face in the control mesh. Each patch is n-sided if it
is locally parameterized over a n-sided face. Analytic expressions for each of
these patches can be derived even in the presencie of extraordinary vertices in
the control mesh, and hence an expression of the limit surface can be obtained.
2. Once an expression of the limit surface is obtained, the dynamic framework can
be developed by treating control vertex positions as a function of time. The
corresponding motion equation can be derived.

102
3. Each patch in the limit surface is treated as a finite element in implementation.
The elemental mass, damping and stiffness matrices along with the general¬
ized force vector can be obtained by either analytic.1 or numerical integration.
Alternatively, the control mesh can be subdivided j times to obtain an approxi¬
mation of the smooth limit surface, and a spring-mass system can be developed
on this approximation mesh. The physical matrices of this system provide an
approximation to the original physical matrices and works well in practice.
5.5 A General Outline of the Framework for Interpolatory Subdivision Schemes
Most of the interpolatory subdivision schemes are obtained by modifying the
butterfly subdivision scheme [30]. Therefore, the framework developed for the modi¬
fied butterfly subdivision scheme in Chapter 4 is pretty general and can be used for
other interpolatory subdivision schemes as well. The only difference is that the basis
functions as well as the set of control vertices for a patch in the limit surface depend
on the chosen interpolatory subdivision rules. It may also be noted that unlike the
approximating schemes, the physical matrices can not be obtained analytically as
the basis functions corresponding to interpolatory subdivision schemes do not have
any analytic.1 expressions in general. Even though these matrices can be obtained via
numerical integration, the spring-mass system developed in Chapter 4 is preferred
for implementation purposes due to efficiency reasons.

CHAPTER 6
MULTIRESOLUTIOX DYNAMICS
The dynamic.' framework developed for subdivision surfaces allows the users to
directly manipulate the smooth limit surface by applying synthesized forces. In any
subdivision scheme, an initial (control) mesh is refined using specific subdivision
rules to obtain a smooth surface in the limit. To develop the dynamic.' framework,
this smooth limit surface is parameterized over the domain defined by the initial mesh
and hence, the limit surface can be expressed as a function of the initial mesh. The
users apply synthesized forces on the smooth limit surface which is transferred back
on the initial mesh via a transformation matrix. The initial mesh evolves over time in
response to the applied forces, and consequently the smooth limit surface deforms to
obtain an equilibrium position where all forces are balanced. As noted earlier, even
though various types of forces can be applied on the limit surface to obtain a desired
effect, the possible shapes that can be obtained via evolution is directly related to
the number of control vertices present in the initial mesh.
The concept of multilevel dynamic's was introduced earlier in order to have vary¬
ing number of control vertices in the initial mesh, and a wide range of shapes with
arbitrary topology can be modeled within the dynamic.' framework. The reader is
103

104
black box to the user
Figuro 6.1. Schematic block diagram of the multilevel dynamic's approach.

105
referred to Section 3.2.5 and Section 3.3.6 of this dissertation for the details of multi¬
level dynamics in the context of dynamic Catmull-Clark surfaces, and to Section 4.2.3
for the dynamic.' modified butterfly subdivision scheme. The concept of multilevel dy¬
namics is illustrated with block diagrams in Fig.6.1. This multilevel dynamics can
be considered as a subdivision surface-based eoarse-to-fine modeling framework and
can be summarized as follows. The user starts with a smooth limit surface which has
a very simple initial mesh with few control vertices. This smooth surface is deformed
by applying synthesized forces so that the limit surface at equilibrium would look
like the final desired shape but without the details in it. Then, the number of control
vertices in the initial mesh is increased by replacing the initial mesh at equilibrium
by another one obtained via one subdivision step on the old initial mesh, and the
old initial mesh is discarded. Both of these old and new initial mesh represent the
same limit surface, but the latter has more control vertices. Xow, this new initial
mesh, and consequently the smooth limit surface, can be deformed to obtain a shape
at equilibrium which has more details. This process is repeated till the deformed
smooth surface has all the desired details.
The limitation of the multilevel dynamic's approach is that once a finer resolution
is chosen, the modeler can not opt for a lower resolution as the dynamic's is defined on
the current initial mesh. There is no way of having the dynamic's defined on a simpler
initial mesh, at the same time preserving the already obtained details in the model
within the multilevel dynamic's approach. This is potentially a severe restriction. For
example, let us imagine the scenario depicted in Fig.6.2. The modeler has recovered

106
(a) Shape recovery
ca Q O
initial shape recovered global shape
with few degrees of
freedom
recovered detailed shape
with large degrees of
freedom
(b) Editing the recovered shape : Global bending
Multilevel dynamics approach
formulate dynamics on the
current control mesh which
has large degrees of freedom,
bend the entire detailed shape.
(time consuming, difficult)
Multiresolution dynamics approach
formulate dynamics on a
coarser level of the control
mesh with few degrees of
freedom, bend the global
shape, and then add back
the details.
(fast, easy)
+ details!
(static)1
Figuro 6.2. Multilevel dynamics vs. multiresolution dynamics.

107
a long straight pipe-like shape with lots of detail by synthesized force application
on a smooth limit surface. This smooth limit surface initially had a simple initial
mesh with few vertices, but the current initial mesh after many model subdivision
steps has large number of control vertices. Xow, if the modeler wants to bend this
straight pipe-like shape, the synthesized forces applied on the smooth limit surface
need to deform an initial mesh with a large number of control vertices. It would have
been much easier and faster if a lower resolution version of the current initial mesh
could be deformed with the details being added back when the shape has bent to
the desired extent in response to the applied synthesized forces. Also, it has been
pointed out in Section 2.5 that the dynamic framework can integrate shape recovery
and shape modeling in a unified framework where the recovered shape can easily be
edited as well. However, if the modeler wants a global change in the recovered shape,
a lower resolution control mesh (than the one obtained via shape recovery process)
may be required, and the details need to be preserved at the same time.
A multiresolution representation of the initial mesh is proposed in this chapter
to solve the above-mentioned problem. At equilibrium, an initial mesh is subdivided
to obtain an initial mesh with more control vertices (degrees of freedom). Instead
of developing the dynamic's on this newly obtained initial mesh as in the multilevel
dynamic's approach, the dynamic's is developed on a representation which views the
newly obtained initial mesh as the previous initial mesh plus detail (wavelet) co¬
efficients needed to get the current initial mesh (see Fig.6.3). This is essentially
developing the dynamic's on a multiresolution representation of the evolving initial

108
( 3 control vertices) ( 3 control vertices, 3 detail coefficients)
(b) Multiresolution approach
Figure 6.3. Representation of the degrees of freedom in multilevel dynamic's and
multiresolution dynamic's approach.

109
mesh which has undergone model subdivision step(s) to increase the degrees of free¬
dom. It may be noted that both the multilevel and the multiresolution approach
have same degrees of freedom to represent the initial mesh obtained after a model
subdivision step, but the degrees of freedom for the multilevel dynamic's approach are
the control vertex positions in the newly obtained mesh whereas, the degrees of free¬
dom for the current approach are the control vertex positions in the previous initial
mesh and the detailed coefficients necessary to obtain the current initial mesh from
the previous one. The multiresolution approach has the benefit that the modeler can
opt for a lower resolution initial mesh later, on which the dynamic's can be defined,
at the same time preserving the detailed coefficients of the initial mesh on which the
dynamic's is currently defined. This concept will be detailed further in later sections,
but first an overview of the multiresolution analysis is provided so that the reader
can easily follow the rest of the material presented in this chapter.
6.1 Overview of Multiresolution Analysis and Wavelets
The basic.' idea of multiresolution analysis is to decompose a complicated function
into a lower resolution version along with a set of detailed coefficients (also known
as wavelet coefficients) necessary to recover the original function. Multiresolution
analysis and wavelets have widespread applications in diverse areas like signal analysis
[55, 81], image processing [26, 56], physics [2], numerical analysis [7], computen- vision
[48] and computen- graphic's [6, 13, 16, 17, 31, 32, 35, 45, 50, 53, 67, 68, 70, 85,
86, 104, 110]. The specific areas in computer graphics that involve multiresolution
analysis and wavelets include, but not limited to, global illumination [16, 17, 35, 85],

110
curvo, mesh and imago editing [6, 32, 110], video processing [31], surface viewing [13],
surface interpolation [70], creating surfaces from contours [67, 68], animation control
[50], polyhedral compression [53], modeling [45, 104] and representing functions on
a sphere [86]. Various applications of wavelets in computer graphic's are detailed in
Stollnitz ot al. [92], The idea of multircsolution analysis is explained next with a
simple example.
Let p-; be a vector containing functional values of some arbitrary function at n
discrete points. A lower resolution representation of the function using m (m < n)
functional values can be obtained from pJ by doing certain weighted averages of the
n functional values stored in pT This is essentially a low pass filtering followed by
down-sampling, and the entire operation can be expressed as
pJ'-1 = AJpJ, (6.1)
where pJ'-1 is the 7?t-valued vector representing a lower resolution version and the
matrix AJ is of size (m, n) and implements the low pass filter along with the down-
sampler.
The lower resolution pJ'-1 has fewer functional values than p7. and hence some
amount of detail present in p7 is lost due to the low pass filtering process. If the low
pass filter A7 is chosen appropriately, it is possible to capture the lost detail by high
pass filtering followed by a down-sampling of the original functional values stored in

Ill
p7. This operation can be expressed as
qJ_1 = B7p7, (6.2)
where q7_1 is a (n — m)-valued vector representing detailed coefficients necessary to
recover p7 from p7_1. The matrix B7 is of size (n — m, n) and implements the high
pass filter along with the down-sampler. The high pass filter B7 is related to the low
pass filter AT These two filters, A7 and B7. are known as analysis filters and the
process of splitting a high resolution vector p7 into a low resolution vector pJ_1 and
a vector qJ_1 storing detail coefficients is known as decomposition.
The original vector p7 can be reconstructed from the vectors pJ_1 and qJ_1 if
the synthesis filters are chosen correctly. This is called the reconstruction process,
which can be expressed as
pi _ pipi 1 + Qlql 1, (6.3)
where PJ is a refinement matrix of size (n. m) and QJ is a perturbation matrix of
size (n.ni — n). The refinement matrix P; encodes the rules for obtaining a vector
of larger size from a given vector and the perturbation matrix QJ encodes how to
obtain a perturbation of (pJ — PJpJ-1) from the given detailed coefficient vector qJ_1.
The matrices P; and Q; are collectively (.'ailed synthesis filters, and are related to
the analysis filters.

112
Figuro 6.4. The filter bank.
The process of decomposition can be continued recursively. For example, p7 is
decomposed into a lower resolution part p7_1 and a detail part q7-1. Then, p7_1
can be expressed as a lower resolution part p7-2 and another detail part q7-2. This
recursive process is depicted in Fig.6.4 and is known as filter bank. The original vector
is decomposed into a hierarchy of lower resolution parts p7_1. p7-2..... p° and detail
parts q7_1. q7-2..... q°. The original vector p7 can be recovered from the sequence
P°: q°: q1: • • •: qJ'-2: qJ_1. This sequence has the same size as that of the original
vector and is known as the wa.velet transform, of the original vector.
The analysis and synthesis filters are designed in such a fashion that the low
resolution versions are good approximations of the original function in a least square
sense. The decomposition and reconstruction processes should have time complexity
0(n), where n is the size of the vector being decomposed or reconstructed. A de¬
tail (wavelet) coefficient value should also be related to the error introduced in the
approximation when that particular coefficient is set to zero.

113
Multiresolution analysis is a framework for developing these analysis and syn¬
thesis filters. It involves derivation of a sequence of nested linear spaces Vo C V1 C
V2 C ... such that the resolution of the functions contained in some space Vj in¬
creases with increasing j. Also, there exists an orthogonal complement space 11 1
for each \ 1. j = 1. 2.... such that V7'-1 © 11 1 = \ i.e.. the linear space V'J_1
and its orthogonal complement space 11 1 together span the linear space Vj. The
inner product between any two functions at some level j. j = 0,1. 2,... needs to be
defined in order to derive the orthogonal complement space.
Let 4>[. 4>2, ..., be a set of basis functions spanning the linear space Vj.
These basis functions are also (.'ailed scaling functions for level j. Xow, V'J-1 C Vj
and hence the basis functions |_\ can be expressed as a linear combination of the basis functions than span the linear
space Vj. This refinability of scaling functions is used to construct the refinement
matrix PJ, which should satisfy the expression
1 = &Pj, (6.4)
where (fiff1,.... & = [(fr[, (fy,..., (fy], and PJ is the (n, m) re¬
finement matrix introduced earlier (Eqn.6.3). Similarly, let L^-1. ..., be
a set of basis functions that span the linear space 11 1. These basis functions, also
known as wavelets at level j — 1, together with the basis functions of linear space
V'i-1 form a basis for the linear space Vj. Next, the perturbation matrix Q; can be

114
constructed such that it satisfies the expression
yj-i = qjqj (6.5)
where fit7 1 = [Ap1, A^1, ■ ■ ■, ’tpn-m], and Q; is the (n, n — m) perturbation matrix
introduced in Eqn.6.3. A more compact expression may be obtained by combining
Eqn.6.4 and 6.5, which can be written as
[&-1 | = & [PJ | QJ]. (6.6)
The analysis filters AJ and BJ should satisfy the inverse relation
[&-1 |
AJ
W
(6.7)
Both the matrices [PJ | QJ] and
are of size (n, n) and satisfy the relation
[PJ I QJ]
(6.8)
6.2 Multiresolution Analysis for Surfaces of Arbitrary Topology
Classically, the multiresolution analysis was developed on infinite domains such
as real line 5ft and plane 5ft2 [22, 23]. Infinite domains are spatially invariant and
hence a single basis function at the coarsest resolution can be dilated and translated
to form the basis functions at higher resolutions. However, many applications need
multiresolution analysis on bounded intervals and techniques for imposing boundary

115
constrains have also boon dorivod [19, 79]. Tho idoa of generalizing multirosolution
analysis on arbitrary manifolds was first introduced by Lounsbory ot al. [52, 53], and
was further improved by Stollnitz ot al. [92] using tho concepts of lifting scheme [93].
In this dissertation, tho multirosolution schemes dorivod by Stollnitz ot al. [92] for
surfaces of arbitrary topology will bo adopted to develop tho multirosolution dynamic.1
framework for subdivision surfaces.
Multirosolution analysis using wavelets can bo categorized into throe typos -
orthogonal [22, 55], somiorthogonal [18] and biorthogonal [20, 93]. A multirosolution
scheme has orthogonal wavelets if tho wavelets are orthogonal to one another, and
every wavelet is orthogonal to every coarser scaling function. It is difficult to con¬
struct orthogonal wavelets that have local support (i.o., non-zero over a small section
of tho domain). Locally supported wavelets are important as they load to sparse
synthesis and analysis filters which in turn load to linear time complexity in vector
size. Wavelets with local support can bo constructed by relaxing tho orthogonality
constraint. If tho wavelets are orthogonal to each other within one resolution but
not across different resolutions, then they are (.'ailed somiorthogonal wavelets. In
a biorthogonal sotting, tho orthogonality constraint is dropped whore 11 1 is some
(and not orthogonal) complement of V'J_1 in Vj. Tho wavelets on arbitrary manifolds
developed by Stollnitz ot al. [92] and used in this dissertation are biorthogonal. Tho
multirosolution analysis on arbitrary manifolds involve (1) developing nested spaces
using subdivision sellemos, (2) selecting an inner product on arbitrary manifold and
(3) constructing biorthogonal wavelets, which are discussed in tho rest of this section.

116
It may bo noted that there is a slight deviation from the standard notational
convention of writing vectors in bold face lower case and matrices in bold face upper
ease letters in a particular situation in the rest of this chapter. The collection of
control point positions are assumed to bo a throe-column matrix (one for each of
the x, y and z components) instead of a vector whoso size is thrice the number of
control points. Consequently, the three-row matrix representing the collection of
basis functions becomes a row vector. This approach is more natural for developing
the wavelet expressions, whereas the opposite was true for the theory developed in
earlier chapters. However, bold face lower case letters are used to represent the three-
column matrix of control vertex positions and bold face upper case letters are used to
represent the vector of basis functions, so that the reader does not get confused with
a new notation for representing the same thing in a different format. The notational
purity is given away to a small extent for the sake of clarity. All the expressions can be
reformulated by merging the columns of the three-column matrix into a single vector,
but the resulting expressions will unnecessarily obscure the intuitiveness contained
in the current set of expressions.
6.2.1 Nested Spaces through Subdivision
In previous chapters, it has been shown that the smooth limit surface generated
by a recursive subdivision scheme in general can be written as
s(x) = J(x)p,
(6.9)

117
where s is the smooth limit surface. J is a collection of basis functions spanning
the space defined by the control mesh S, p is the vertex positions in the control
mesh and x e S. A general outline on how to obtain this expression for any type of
subdivision scheme has been provided in Section 5.4 and 5.5 of this dissertation and
specific examples can be found in Eqn.3.15 and Eqn.5.5 for Catmull-Clark subdivision
scheme and in Eqn.4.11 for the modified butterfly scheme.
Let the control mesh defining the smooth limit surface be S°. Renaming the
control vertex vector p to p° and the basis (sealing) function matrix J to J°. Eqn.6.9
can be rewritten as
s(x) = J°(x)p°, (6.10)
where x e 5°. Let S1 be a mesh obtained by subdividing SQ once using the sub¬
division rules. If the control vertices of the mesh S1 are collected in a vector p1,
then
P1 = P^, (6.H)
where P1 is the subdivision (refinement) matrix. However, the control mesh S1 has
the same limit surface (as it is obtained via one subdivision step on the control mesh
S°) and hence the same limit surface can be written as
s(x) = J^p1
J^xJPV
(6.12)

118
where J1 is a collection of basis (scaling) functions spanning the space defined by S1.
From Eqn.6.10 and Eqn.6.12, it is obvious that
J°(x) = J^xJP1, (6.13)
which essentially means that the scaling functions at subdivision level 0 can be ex¬
pressed as a linear combination of the sealing functions at subdivision level 1. i.e..
Vo (S°) C y1(5'°). This can be generalized for other subdivision levels as well, and
hence subdivision naturally leads to a nested space sequence Vo(S°) C y1(5'°) C
V2(S°) C ... which will be utilized for multiresolution analysis on arbitrary mani¬
folds.
6.2.2 Inner Product
The inner product proposed in Stollnitz et al. [92] is used in this dissertation.
If fa and ft, are two functions defined on the domain SQ. then their inner product
< fa, fb > is given by
= Y “TT Í (6-14)
jeF(s°) Jxe'r
where F(S°) is the set of (triangular or quadrilateral, depending on the subdivision
scheme) faces in the control mesh S°, 7 is a face in the set F(S°), 0.(7) is the area of
the face 7 and dtx. is the usual differential area in 5ft3.

119
Figuro 6.5. Subdivision refinement of a tetrahedron.
6.2.3 Biorthogonal Surface Wavelets on Arbitrary Manifold
The biorthogonal wavelets discussed hero can be developed for any subdivision
scheme in general. However, some minor modifications are needed in some situations.
The construction is based on the lifting scheme [93] whore first “lazy” wavelets at
level j — 1 are constructed using some scaling functions at level j. and then those
wavelets are “lifted” so that they become as orthogonal as desired to the scaling
functions at level j — 1. Those concepts are described below using a simple example.
A more detailed discussion on this construction can be found in Stollnitz ot al. [92],
Let S°, as shown in Fig.6.5, bo a (tetrahedral) control mesh which when refined
using some subdivision rules loads to a smooth surface in the limit. The subdivision
rulos refine each triangular face into four triangular faces by introducing now vertices
corresponding to each edge in the mesh as shown in Fig.6.5. Let the smooth surface
in the limit bo s (denoted by S°° in Fig.6.5). Let p° be the collection of four control
vortex positions defining the control mesh S°. Xow, there exists scaling functions
f(x), x e S°, associated with each vortex p», * = 4, such that they span the

120
space Vo(5°). Those scaling functions depend on the subdivision scheme involved
and may or may not have analytic.1 expressions. Those scaling functions are collected
in J°(x) = [5i(x), 2(x)- 5l(x): 5l(x)]- The smooth limit surface s can be written
as
s = J°(x) p°, (6.15)
whore x e 5°. The control mesh 51 is obtained by subdividing the control mesh 5°
once. The ten control vertices of the mesh 51 can be categorized into two classes :
(1) four control vortices corresponding to the four control vortices in original mesh
5° (the filled dots in Fig.6.5) and (2) six control vortices corresponding to the six
edges in original mesh 5° (the unfilled dots in Fig.6.5). If the control vortex positions
of the mesh 51 are collected in p1 such that the “old” vortices corresponding to the
vortices in the original mesh procedo the “now” vortices corresponding to the edges
in the original mesh, then
p°, (6.16)
whore P1 is the subdivision matrix, and P*M and P^ are portions of the subdivision
matrix that encode rules to obtain the “old” and “now” vortices respectively. Basis
functions corresponding to the vortices in p1 can bo collected in J1(;r) = [Jb,d(;r) |
Jjje?i)(;r)] = [5Í(X); • • •; 5i(x) | 5(x): • • •, 5Ío(x)L whore the first four basis func¬
i_o
P*p
P1
old
tions are associated with old vortices and the last six basis functions are associated

121
with now vortices. The samo smooth limit surface s can bo written as
s = J^x) p1. (6.17)
Evidently, the dimension of Vo (S°) and \n(S°) are 4 and 10 respectively. In the
lazy wavelet construction, a complement space W° (S°) of dimension 6 is constructed
with the set of basis functions corresponding to the now vertices in S1, i.e.. the
wavelets spanning IE0(S'0) are W°(x) = [A°(x),.... r°(x)’ = [J0(x)].
This construction is called “lazy” because no extra computation need to be done to
derive the wavelets. The refinement relations for the lazy wavelet construction can
be compactly written as
[J°(x) | W°(x)] = [JÍW(.r) | [P¡azy I Qlazyh
(6.18)
whore P* = P1 and
[pLy I Qia*»]
o
P1 I
new
0 and I denoting zero and identity matrix respectively. After developing the synthesis
filters for the lazy wavelets, the analysis filters can be derived using the inverse relation

122
and aro given by
-i
A1 '
lazy
P1
aid
0
0
R1
lazy.
P1
new
i
_pi fpi \_1
naw\ old)
I
(6.19)
The lazy wavelet construction developed hero can bo generalized for any subdivision
level j in a straight-forward manner. The synthesis and analysis filters at level j for
a subdivision scheme can bo written as
[P
i
lazy
pj
r old
0
pj X
new
(6.20)
and
-i
' Aj
lazy
pi
r old
0
(p^r1
0
RJ
lazy
pi
new
i
I
respectively, whore PJnld encodes subdivision rules for obtaining “old” vortices at j-
th level corresponding to the vortices in the (j — l)-th level control mesh and PJnew
encodes rules for obtaining the other “now” vortices in the j-th level control mesh.
It may bo noted that the lazy wavelet construction loads to sparse synthesis
matrices. However, the corresponding analysis filters may not bo sparse in general as
inverse of PJnld is not necessarily sparse. This implies that oven though the synthesis
can bo done in linear time with respect to the number of vortices, the complexity of
the analysis depends on how fast the inverse can bo computed. The synthesis can not
bo done in linear time in case of approximating subdivision schemes. However, PJM

123
Wavelet associated with the edge
connecting a and b
Lazy : the scaling function associated with
the introduced vertex at higher level
Lifted : the scaling function associated with
the introduced vertex at higher level
- ( weighted combination of the scaling
functions associated with the solid
vertices)
Figuro 6.6. Wavelet construction.
is an identity matrix for interpolatory subdivision schemes, and hence both synthesis
and analysis can be done in linear time for interpolatory subdivision schemes. The
simplified expressions of the synthesis and analysis filters in case of interpolatory
subdivision scheme are given by
[P
j
lazy
0
pj X
new
(6.22)
and
Aj
lazy
RJ
lazy
new
(6.23)
respectively.

124
The lazy wavelets are easy to construct. However, a lower resolution mesh
obtained using the analysis filters of the lazy wavelet construction is far from the
least-squares-best approximation. Lazy wavelets at some level (j — 1) are nowhere or¬
thogonal to V^'-1. The lifting scheme [93] improves the lazy wavelets at level (j — 1) by
subtracting a liner combination of the coarser level scaling functions which overlap to
a given wavelet. The resulting wavelet function becomes somewhat more orthogonal
to the coarser level scaling functions than the original lazy wavelet. Mathematically,
a “lifted” wavelet at level (j — 1) can be expressed as
'•’/'¿W = Mazy A*) - XXi<#_1(x)
l
= 'W- (6-24)
l
where ?!-th lazy wavelet function at level (j — 1) is the scaling function associated
with the ?'-th “new” vertex at level j, and l is restricted to a few vertices at level
(j — 1) in the neighborhood of the ?'-th new vertex at level j (Fig.6.6). In order to get
the values of sJu for the lifted wavelet i, a linear system of equations is derived and
solved in a least squares sense by imposing the constraints < (x)-O'*. '(x) > =
0 for all k, such that the supports of ¡.’¿j/^x) and non-zero entries is in sj - needs to be controlled. A large number of non-zero entries
will lead to a lifted wavelet function that is nearly orthogonal to the coarser level
scaling functions, at the same time the multiresolution analysis will become more
time consuming, and hence a tradeoff needs to be decided as far as orthogonality
and time of computation is concerned. In Stollnitz et al. [92], “k-disk wavelets” are

125
constructed for triangular mesh-based subdivision schemes where the non-zero entries
of sjj are restricted to the vertices at level (j — 1) which are within the ^-neighborhood
of the ?'-th “new” vertex at level j. The reader is referred to Stollnitz ot al. [92] for
the details of the construction. Once the non-zero values of sjA are computed, they
can be assembled to a matrix ST Then, the expressions for the synthesis and analysis
filters of the lifted wavelet scheme can be written as
rpi i ry
l lazy I Hazy
(6.25)
and
' A3
ft
pi
_Hfl
respectively.
6.3 Multiresolution Representation
The multiresolution analysis on arbitrary manifold was developed to obtain good
low resolution approximations of complicated meshes. This is a top-down technique
where one starts with a complicated arbitrary topology mesh and goes on obtain¬
ing lower resolution meshes using analysis filters. The complicated mesh on which
multiresolution analysis is applied can be reconstructed using the synthesis filters.
In this dissertation, the multiresolution technique is used in a novel bottom-up ap¬
proach where an evolving control mesh of a dynamic subdivision surface is subdivided
at equilibrium to obtain more degrees of freedom, and then, the resulting control mesh
is treated as the previous control mesh and a collection of detail (wavelet) coefficients.
A J i C./p./
lazy ' 0 ^lazy
pi
lazy
(6.26)

126
Of course, the detail coefficients are zero when an control mesh is replaced by another
control mesh obtained by a pure subdivision step on the previous mesh, but becomes
non-zero as the new control mesh (represented as previous control mesh and wavelet
coefficients) evolves over time in response with the synthesized force application. This
idea is further illustrated in the rest of this section.
Let s be a smooth limit surface obtained via infinite number of subdivision
steps on an initial mesh S°, whose vertex positions are collected in p°. As mentioned
earlier, the limit surface can be written as
s = J°p°, (6.27)
where J° is a collection of basis functions. If the control mesh S1 is obtained by
subdividing SQ once, then the same limit surface can be expressed as
s = .^p1, (6.28)
where p1 is the collection of vertex positions of the mesh S1 and J1 is the collection
of corresponding basis functions. Since p1 = P1p°, the limit surface s can also be
written as
s = J^p0. (6.29)
The difference between Eqn.6.27 and Eqn.6.28 is that the limit surface s has more
degrees of freedom (control vertices) for representation in the latter in comparison
with the former. However, in Eqn.6.29, even though the limit surface is expressed

127
using the samo basis functions as that of S1, the degrees of freedom remains the same
as that of S°. Eqn.6.29 can be modified in the following manner so that it uses same
basis functions and same number of degrees of freedom as that of S1.
s = Jx [P1 Q1]
whore q° is the collection of wavelet coefficients whoso values are zero, P1 and Q1
are the synthesis matrices andQ^y, respectively. The above formulation yields
a multiresolution representation of the control mesh S1 obtained via one subdivision
stop on the previous control mesh S°. The mesh S1 is represented as a mesh at level
0 along with the detail (wavelet) coefficients at level 0. As mentioned earlier, those
coefficients would become non-zero when the smooth limit surface deforms over time
duo to synthesized force application. Similarly after one more subdivision stop, the
now control mesh S2 can be viewed as control vortex positions at level 0, wavelet
coefficients at level 0 and wavelet coefficients at level 1, and can be written as
(6.30)
s
J2 p2
J2 [P2 Q2]
J2 [P2 Q2]
P1 Q1 0
0 0 1
(6.31)

128
where J2 is the collection of basis functions at level 2, p2, p1 and p° are the control
vertex positions at level 2, 1 and 0 respectively, q1 and q° are the wavelet coefficients
at level 1 and 0 respectively. I is the identity matrix and 0 is the zero matrix. This
idea can be generalized for a control mesh obtained after j-th subdivision and the
corresponding expression can be written as
JJ AJ pi
r'j
(6.32)
where (p2)1 = [p°q° ... q2 2q2 1]. and
A2 = [P2 Q2
1
1
1
o
7
O’
1
1
to
Q2-2 0
1
o
l
O
O
O’
0 .
1
O
I
o
0 I
I
o
0 I
i
h—H
I
o
0
I .
i
h—H
The multiresolution dynamic's is developed using this formulation in the next
section.
6.4 Dynamics
The smooth limit surface s can be made dynamic if the control vertex positions
at level 0 and wavelet coefficients at level 1,2,.... (j — 1) are functions of time. The
velocity of the smooth surface, controlled by the mesh Sj obtained through j steps
of model subdivision, is given by
s(x. p2) = J2(x)A2p2,
(6.33)

129
where the overstruck dot denotes the time derivative and x e Si The Lagrangian
motion equation (refer to Eqn.2.5) can be derived in a similar fashion as mentioned
in earlier chapters, and is given by
M>'p¿ + D-'p-' + K-'p? = fj>, (6.34)
where MJ, D; and K; are the j-th level mass, damping and stiffness matrices respec ¬
tively and is the generalized force vector at level j. If /.¿(x) is the mass density of
the subdivision surface model, then the mass matrix at j-th level is given by
M> = [ fi{x){AJf{JJ{x))TJJ{x)AJdx.
The damping matrix DJ can be derived in a similar fashion, while the stiffness ma¬
trix K; can be obtained following the techniques described in Section 4.3.2. The
generalized force vector at level j can be written as
Pp = f (AJ'f (JJ'(x)f f(x.t)dx.
6.5 Implementation Details
Multiresolution representation of the evolving control mesh achieves eoarse-to-
fine as well as fine-to-coarse manipulation of the smooth limit surface. For example,
the user starts with a deformable smooth surface with a simple control mesh S°
and directly manipulates the limit surface by applying synthesized forces. When an

130
equilibrium is obtained, the user can increase the degrees of freedom by switching to
a control mesh S1 which is theoretically obtained by applying one subdivision step
on S°, but represented as a collection of vertex positions in S° along with wavelet
coefficients at level 0 needed to reconstruct S1. The wavelet coefficients are zero at
the point of switching, but becomes non-zero as the control mesh S1, represented
as mentioned above, evolves over time. The user can further increase the degrees
of freedom to obtain more localized effect by introducing control mesh S2 which is
represented as control vertex positions of mesh S°, along with wavelet coefficients
at level 0 and 1. Thus the user can have coarse-to-fine manipulation of the smooth
limit surface. This facility is also present in the multilevel dynamic's approach as
mentioned earlier. However, the multilevel dynamic's approach does not support fine-
to-eoarse manipulation - it fails if the user wants to apply synthesized forces on a
coarser control mesh at level j after moving up to level j + k. This can be achieved
in multiresolution approach by simply using the dynamic.' equation of motion at level
j. and applying synthesized forces on a smooth limit surface which is obtained by
multiresolution synthesis using “time varying” control vertex positions at level 0
and wavelet coefficients at level 0,1,..., (j — 1) and “static” wavelet coefficients at
level j, (j + 1),..., (j + k — 1). If the user switches back to level (j + k), then
the control vertex positions at level 0 as well as all the wavelet coefficients at level
0,1,..., (j — 1), j,(j + 1),..., (j + k — 1) become function of time, and the system
switches back to the motion equation of level (j + k).

131
The actual implementation differs slightly from the formulation to achieve effi¬
ciency in coarse-to-fine and fine-to-eoarse manipulation. In the implemented system,
the user starts out with a smooth limit surface which has a simple control mesh with
very few degrees of freedom. The model grows and when the equilibrium is achieved,
one step of subdivision yields larger degrees of freedom. This larger degrees of free¬
dom are the control vertex positions in this finer resolution and not control vertex
positions in lower resolution along with wavelet coefficients in that lower resolution.
The model can grow further, and if more subdivision step is needed to increase the
degrees of freedom, it is handled in a similar fashion. If at any certain point of time
the user needs to go back to a coarser level control mesh, wavelet decomposition is
done on the higher resolution base mesh. This decomposition yields vertex positions
in the coarser level mesh (which becomes the new base mesh) and non-zero wavelet
coefficients. The lower resolution mesh evolves with time due to the force applied on
the limit surface. It may be noted that the limit surface in this scenario is obtained by
doing wavelet reconstruction with non-zero wavelet coefficients at some levels. The
user can choose further coarser level control mesh by repeating the same process.
In the implemented system, the user can achieve global deformation on a de¬
tailed mesh by going down couple of levels using wavelet decomposition, retaining
the wavelet coefficients, and applying force on the limit surface (obtained through
wavelet reconstruction process) and letting the coarser level mesh evolve. Similarly,
to achieve local deformation on a coarser mesh, the user can move up couple of steps

132
by doing subdivision (and adding non-zero wavelet coefficients at higher levels, if
present) and applying synthesized forces in the region of interest.
The formulation using wavelet representation has mathematical niceties, but
inefficient for implementation purposes. The synthesized force is applied on the limit
surface. To evaluate the limit surface, wavelet reconstruction needs to be done at each
time step starting from the original control mesh upto a specified level if an explicit
wavelet representation of the evolving control mesh is maintained as mentioned in
the formulation. On the other hand, if the evolving mesh is control vertex positions
at that resolution, then the wavelet reconstruction starts from that resolution for
evaluating the limit surface. Wavelet decomposition needs to be done while going
down from finer to coarser resolution, but this happens only once in a while, and not
at each time step. Therefore, the implemented version is much more efficient. It may
be noted that the motion equation at any level is implemented using finite element
techniques discussed in earlier chapters for various subdivision schemes.

CHAPTER 7
SYSTEM ARCHITECTURE
The architecture of the prototype system is described in this chapter. The basic.1
building blocks of the system are different modules performing different tasks. The
overall architecture of the system is shown in Fig. 7.1.
The user first provides the input about the geometry and topology of the initial
mesh. Typically these information consist of some type of connectivity information
and control vertex positions in 3D. The topological information processing module
derives other necessary topological information from the given input data, and passes
all the information to the subdivision module. The subdivision module performs pre¬
specified number of subdivision steps on the initial mesh, and then the finite element
analysis module computes different elemental matrices needed by the update engine.
The user can apply different types of synthesized forces on the limit surface, and
the model is constantly updated to obtain an equilibrium position. The updated
information is passed to the display module which renders the deforming smooth
surface on the screen. The user is also allowed to choose different model parameter
values like mass density, damping density, spring constants etc. Details of the specific.1
modules are provided in the rest of this chapter.
133

134
r
Figuro 7.1. System architecture.

135
7.1Topological Information Processing Modulo
In the implemented system, the user needs to provide the connectivity of the
initial mesh by listing the vertices of every face. The vertex information of a face
must be provided in an orderly fashion (either clockwise or anti-clockwise). The
topological information processing module derives other necessary information like
vertex connectivity information of the edges, edge connectivity information of the
faces etc. from the input provided by the user.
7.2Subdivision Module
This module applies different subdivision rules to refine an input control mesh.
Currently, Catmull-Clark and modified butterfly subdivision schemes are supported
in the system. Other subdivision schemes can also be easily incorporated. The
subdivision scheme for a specific application is chosen by the user. However, there
are certain inherent restrictions depending on the chosen subdivision scheme. For
example, butterfly subdivision scheme can be chosen only if the initial (control)
mesh has triangular faces. Once the user provides input about the initial mesh
and the chosen subdivision scheme, this module then subdivides the initial mesh
a pro-specified number of times to obtain a high resolution mesh which effectively
approximates the smooth limit surface.
7.3Finite Element Analysis Modulo
The finite element analysis modulo determines the typo of finite element to
bo used and the total number of elements describing the smooth limit surface. It
computes all elemental matrices such as mass, damping and stiffness matrices, as

136
well as determines the set of vertices in the initial mesh controlling each individual
element. This module also provides the information about the global mass, damping
and stiffness matrices without actually assembling them from individual elemental
matrices.
7.4 Force Synthesis Module
The user can apply various types of forces on the smooth limit surface. Currently
spring forces, balloon forces, and image gradient-based forces are being supported.
Other types of forces can also be easily incorporated. The user applies spring forces
on the limit surface by providing points in 3D from which the springs need to be
attached. The 3D point specification is done either interactively from mouse input or
from files. To apply balloon forces, the user needs to specify whether the ballooning
force is to inflate the model or to deflate the model. Volume images are provided
as input to apply image gradient-based forces. A detailed discussion on these force
application methods was provided in Section 3.3.4.
7.5 Update Engine
After the user provides all the necessary information about the model and the
force application method, the update engine constantly updates the dynamic sub¬
division surface model to achieve an equilibrium position. The discretized second
order differential equation of motion leads to solving a large sparse linear system
of equations. Conjugate gradient [34, 75], an iterative linear system solver, is used
for this purpose. The user can choose either multilevel dynamic's or multiresolution

137
dynamic's approach, and the update engine accordingly chooses algorithms when the
degrees of freedom of the evolving surface model is changed.
7.6 Display Module
The update engine constantly updates the dynamic.1 model, and the relevant
geometric.1 and topological information is passed to the display module for rendering
the evolving smooth surface. The display module uses OpenGL graphic's libraries
for rendering purposes. It also serves as the input module for mouse-based user
interactions.

CHAPTER 8
APPLICATIONS
The proposed dynamic.1 framework extends the application areas of subdivision
surfaces beyond the traditional modeling domain. The framework enhances the suit¬
ability of subdivision surfaces for geometric.1 modeling. At the same time, the proposed
dynamic.1 models can be used for recovery of shapes from range and volume data, non-
rigid motion estimation, multiresolution editing and natural terrain modeling. The
applicability of the dynamic.1 model in these areas is further illustrated in the rest of
this chapter.
8.1 Geometric Modeling
The proposed dynamic.1 framework for subdivision surfaces can be used to rep¬
resent a wide variety of arbitrary topological shapes with remarkable ease. In a
typical geometric.1 modeling application using dynamic.1 subdivision surfaces, the user
can specify any mesh as the initial (control) mesh, and the corresponding limit surface
can be sculpted interactively by applying synthesized forces. Any dynamic.1 subdivi¬
sion surface model can be used for the modeling purpose. The examples shown in
this section use dynamic.1 Catmull-Clark and dynamic.1 modified butterfly subdivision
surface models.
138

139
Figuro 8.1. (a), (b). (e) and (d) : Initial shapes (obtained applying Catmull-Clark
subdivision rules on control meshes); (o). (f), (g) and (h) : the corresponding modified
shapes after interactive force application.

140
First, dynamic Catmull-Clark surface models as presented in Chapter 3 are used
for shape modeling. The special elements in the limit surface are darkly shaded and
the normal elements are lightly shaded in all the modeling examples. It may be
noted that the formulation presented in Chapter 5. which will have only one type
of finite element in the limit surface, can be used as well. In Fig.8.1, several initial
surfaces obtained from different control meshes using Catmull-Clark subdivision rules
and the corresponding modified surfaces after interactive spring force application are
shown. To change the shape of an initial surface, springs are attached from different
points in 3D to the nearest points on the limit surface such that the limit surface
deforms towards these points generating the desired shape. In Fig.8.1 (a), an open
surface defined by an initial mesh of 61 vertices and 45 faces is shown. The mesh
has one extraordinary vertex of degree 5. This limit surface is modified by applying
spring forces interactively, and the modified surface is depicted in Fig.8.1(e). A
torus, defined by an initial mesh of 32 vertices and 32 faces, and its modified shape is
shown in Fig.8.1 (b) and (f) respectively. The initial mesh of the smooth limit surface
shown in Fig.8.1(c) has 544 faces and 542 vertices, 8 of them are extraordinary
vertices of degree 5. The limit surface is modified interactively by applying spring
forces from various points in 3D and the modified shape is depicted in Fig.8.1(g).
Note that the extent of deformation has been interactively controlled by varying the
stiffness of the attached springs. The upper portion of the limit surface has been
deformed by applying spring forces of higher magnitude, whereas the lower portion
has been modified by applying spring forces of lower magnitude. The spread of the

141
(o) (f) (g) (10
Figuro 8.2. (a), (b), (e) and (d) : Initial shapes (obtained applying modified butterfly
subdivision rules on control meshes); (e), (f), (g) and (h) : the corresponding modified
shapes after interactive sculpting via force application.
deformation effect is (dearly larger in the latter case for obvious reasons. Finally, a
flat shoot defined by an initial mesh of 64 faces and 81 vortices, shown in Fig.8.1(d),
is deformed interactively to obtain the hat-like shape shown in Fig.8.1(h).
Next, shape modeling examples using the dynamic.' modified butterfly subdivi¬
sion surface model are shown. The limit surface hero consists of a single typo of
smooth triangular finite element patches, irrespective of the number of extraordinary
vortices in the control mesh. In Fig.8.2, several initial surfaces obtained from different

142
control meshes using the modified butterfly subdivision rules and the corresponding
modified surfaces after interactive sculpting are shown. As with the dynamic Catmull-
Clark subdivision surfaces, the user can attach springs from different points in 3D to
the nearest points on the limit surface and the limit surface deforms towards these
points to generate the desired shape. The initial mesh of the smooth surface shown
in Fig.8.2(a) has 125 faces and 76 vertices (degrees of freedom), which is deformed
to the smooth shape shown in Fig.8.2(e) by interactive spring force application. The
initial mesh of the closed cube-like shape in Fig.8.2(b) has 24 faces and 14 vertices.
This cube-like shape is deformed to the shape shown in Fig.8.2(f). The one hole torus
in Fig.8.2(e) and the corresponding modified shape in Fig.8.2(g) have initial meshes
with 64 faces and 32 vertices. A two hole torus with a control mesh of 272 faces
and 134 vertices, shown in Fig.8.2(d), is dynamically sculpted to the shape shown in
Fig.8.2(h).
8.2 Shape Recovery from Range and Volume Data
The dynamic subdivision surface models can recover the underlying shape of
a given range or volume data set in a hierarchical fashion. The initialized model
deforms under the influencie of the synthesized forces from the range or volume data.
When an approximate shape is recovered, a new control mesh can be obtained by
doing one step of subdivision on the initial mesh thereby increasing the degrees of
freedom to represent the same limit surface, and a better fit to the given range data
set can be achieved. It may be noted that the model initialization is interactive,
and the initialized model can have any control mesh of the desired genus. However,

143
(a) (b) (c)
Figuro 8.3. (a) Rango data of a bulb along with tho initialized modol, (b) an in-
tormodiato stago of ovolution, and (e) tho fittod dynamic Catmull-Clark subdivision
surfaco modol.
an initial mosh with fow dogroos of froodom usually porforms bottor in torms of tho
compact roprosontation of tho underlying shape.
Any dynamic subdivision surface modol can be used for shape recovery. Next,
throe examples of shape recovery from range data sots are shown. Tho first two
examples use dynamic.' Catmull-Clark subdivision surfaco modol and tho third one
uses dynamic.' modified butterfly subdivision surfaco modol. Tho error in fit, which
is defined as tho maximum distance between a data point and tho nearest point on
tho limit surfaco as a percentage of tho diameter of tho smallest sphere enclosing tho
object, is approximately 3% in all tho experiments with range data. Tho time of
dynamic.' ovolution for fitting tho range data sots used in tho experiments is approxi¬
mately 3 minutes in a SGI O2 workstation.

144
Figuro 8.4. (a) Rango data of an anvil along with tho initializod modol, (b) an
intormodiato stago of ovolution, and (c) tho fittod dynamic Catmull-Clark subdivision
surface modol.
(a)
(b)
(<0
Figuro 8.5. (a) Rango data of a head along with tho initializod modol, (d) tho fittod
dynamic butterfly subdivision modol, and (c) visualization of the shape from another
view point.

145
The examples shown in Fig.8.3 and Fig.8.4 involve dynamic Catmull-Clark sub¬
division surface. The same shading convention for normal and special elements as
mentioned earlier is used. In both of these examples, the initialized model had 96
faces and 98 vertices. 8 of them being extraordinary vertices of degree 3. The final
fitted model, obtained through one step of subdivision, has a control polygon of 384
faces with 386 vertices. In Fig.8.3, the dynamic Catmull-Clark subdivision surface
model is fitted to laser range data acquired from multiple views of a light bulb. Prior
to applying the model fitting algorithm, the data were transformed into a single refer¬
ence coordinate system. The model was initialized inside the 1000 range data points
on the surface of the bulb. In the next experiment, the dynamic Catmull-Clark sub¬
division surface model is fitted to an anvil data set (Fig.8.4). The data set has 2031
data points. It may be noted that the final shape with 3% error tolerance uses very
few control points for representation in comparison with the number of data points
present in the original range data set.
In the last example with range data set, the shape of a human head is recovered
from a range data set using dynamic modified butterfly subdivision surface model
(Fig.8.5). The initialized model has a control mesh comprising of 24 triangular faces
and 14 vertices whereas the control mesh of the fitted model has 384 triangular
faces and 194 vertices. The range data set has 1779 points in 3D, and a compact
representation of the recovered shape using few degrees of freedom in comparison
with original data set is obtained in this example as well.

146
Figuro 8.6. (a) A slice from a brain MRI, (b) initialized model inside the region of
interest superimposed on the slice, (c) the fitted model superimposed on the slice,
and (d) a 3D view of the dynamic Catmull-Clark subdivision surface model fitted to
the cerebellum.
Figure 8.7. (a) Data points identifying the boundary of the caudate nucleus on a MRI
slice of human brain, (b) data points (from all slices) in 3D along with the initialized
model, and (c) the fitted dynamic.' butterfly subdivision model.

147
The application of the dynamic.' subdivision surface models to anatomical shape
recovery from 3D volumetric.' MRI data is shown in the next two examples. The
first one uses dynamic.' Catmull-Clark model and the next one uses dynamic.' modified
butterfly subdivision surface model.
A dynamic.' Catmull-Clark subdivision surface model is fitted to a cerebellum
(a cortical structure in brain) given an input of 30 sagittal slices from a MR brain
scan. As in the examples with range data, the initialized model had 96 faces and 98
vertices. 8 of them being extraordinary vertices of degree 3. The final fitted model,
obtained through one step of subdivision, has a control polygon of 384 faces with 386
vertices. Fig.8.6(a) depicts a slice from this MRI scan and the model initialization
is shown in Fig.8.6(b). Continuous image based forces are applied to the model and
the model deforms under the influence of these forces until maximum conformation
to the boundaries of the desired cerebellum shape. The final fitted model is shown
in Fig.8.6(c). A 3D view of the fitted model is depicted in Fig.8.6(d).
In the next example, the shape extraction of a caudate nucleus (another cortical
structure in human brain) is presented from 64 MRI slices, each of size (256,256).
Fig.8.7(a) depicts a slice from this MRI scan along with the points placed by an
expert neuroscientist on the boundary of the shape of interest. Fig.8.7(b) depicts the
interactively placed sparse set of data points (placed in some of the slices depicting the
boundary of the shape of interest) in 3D along with the initialized model. Note that
points had to be interactively placed on the boundary of the caudate nucleus in MR
slices lacking image gradients which delineate the caudate from the surrounding tissue

148
in the imago. Continuous imago-based forces as well as spring forces are applied to
the model and the model deforms under the influence of those forces until maximum
conformation to the boundaries of the desired caudate shape. The final fitted model
in 3D is shown in Fig.8.7(e). The initialized dynamic butterfly subdivision surface
model has a control mesh comprising of 24 triangular faces and 14 vertices whereas
the control mesh of the fitted model has 384 triangular faces and 194 vertices.
8.3 Non-rigid Motion Estimation
The dynamic subdivision surface models can be used effectively to estimate the
motion of a non-rigid object from a given time sequence of range or volume data. An
example is shown in Fig.8.8 where the motion of the left-ventricular chamber of a
canine heart is tracked over a complete cardiac cycle using dynamic modified butterfly
subdivision surface model. The data set comprised of 16 3D CT images, with each
volume image having 118 slices of 128 x 128 pixels. First, the shape is recovered from
one data set using image-based (gradient) as well as point-based forces. As before,
the initialized model has a control mesh comprising of 24 triangular faces and 14
vertices whereas the control mesh of the fitted model has 384 triangular faces and
194 vertices. Once the shape is recovered from one data set, this fitted model is
used as an initialization for the next data set to track the shape of interest. The
snapshots from motion tracking are shown in Fig.8.8 for the 16 volume data sets. It
may be noted that the control mesh describing the smooth surfaces shown in Fig.8.8
has only 384 triangular faces with a total of 194 vertices as mentioned earlier. This
experiment clearly shows that our model can be used to track a shape of interest

149
Figuro 8.8. Snapshots from the animation of canino heart motion over a cardiac cycle
using the dynamic butterfly subdivision model.

150
Figuro 8.9. (a) The initialized model along with the data sot; (b) the fitted model
with two subdivisions on the initial mesh, along with attached springs for editing.
The model after editing (c) at lower resolution, (d) at the same resolution of the
fitted model, and (o) at higher resolution.
from a sot of time dependent volume data sots in an efficient manner. Note that no
other existing purely geometric subdivision surface technique can bo used with (time
varying) continuous data sots.
8.4 Multirosolution Editing
The proposed dynamic.' subdivision techniques present hierarchical shape recov¬
ery and shape modeling within a common framework, whore the modeler can scan
in 3D data points of a prototype model, recover the underlying shape from the point

151
sot. and then edit the recovered shape. The multiresolution representation of the
evolving control mesh as presented in Chapter 6 enables the modeler to edit the
smooth limit surface at any desired level. Within the proposed multiresolution dy¬
namic framework, the modeler does not need to build a model from scratch (unlike
other shape modeling techniques), and there is no need of using computationally in¬
tensive remeshing techniques for multiresolution representation (unlike other shape
recovery techniques). The idea is illustrated in Fig.8.9, where the shape of a head is
first recovered from a range data set and then edited at various levels using spring
forces from two points in 3D. The effect was more global in the lowest level edited,
and the effect became increasingly local as the level of editing was increased.
8.5 Natural Terrain Modeling
In this section, a novel technique for synthesizing realistic.1 terrain models from
elevation data is developed by using the dynamic.1 finite element method-based subdi¬
vision surface model for surface reconstruction, and a variant of the successive random
addition method for fractal surface synthesis. Roughness is an essential characteristic
of the natural terrains and hence traditional surface reconstruction methods using
smoothness constraints do not yield the desired results. Fournier et al. [33] first pro¬
posed a random displacement technique for synthesizing fractal surfaces which was
later modified by Saupe [84] in his successive random addition scheme of generating
fractals. Yokoya et al. [107] improved these schemes by adding data constraints.
Szeliski and Terzopoulos [94] proposed constrained fractal surfaces using a Gibbs
sampler algorithm which was later improved by Vomuri et al. [102, 103]. Arakawa

152
and Krotkov [1] refined the original Gibbs sampler technique by redefining the tem¬
perature (control) parameter to obtain better control of roughness in the fitted (con¬
strained) fractal surface. However, the (CPU) execution times reported in their work
are very high, thus making their scheme unattractive for many applications.
All the techniques for natural terrain modeling mentioned above usually needs
a grid of very large size to model realistic.1 terrains, especially with irregularly spaced
data. Riimelin [82] developed an interesting fractal interpolation algorithm whic.ii
can generate interpolating fractal surfaces for regularly or irregularly spaced data.
However, this scheme is computationally intensive for large problems as pointed out
in Voniuri et al. [103].
The subdivision schemes produce a smooth surface in the limit, and hence they
are not suitable for synthesizing natural terrains which are rough. A scheme is devel¬
oped in this dissertation using a variant of the successive random addition method
[84] such that the limit surface looks like “natural” terrain. In the original successive
random addition technique, an equally spaced rectangular grid is refined by inter¬
polating the midpoints of each rectangular cell (thereby dividing each rectangular
cell into four rectangular cells) and then all grid positions are perturbed by addi¬
tion of a Gaussian noise. This process is carried out recursively to obtain a fractal
surface whose roughness is controlled by the variance of the added Gaussian noise
at different refinement levels. In this dissertation, the butterfly subdivision scheme
has been modified, where vertex positions at various levels of subdivision are perturbed

153
(a) (b) (c)
Figuro 8.10. (a) Discreto elevation data sot (4096 points), (b) fitted dynamic butterfly
subdivision surface model with 841 vortices (without noise addition), and (e) fitted
dynamic subdivision surface model with 841 vortices when Gaussian noise is added.
by addition of a Gaussian noise whose va.ria.nee controls the roughness of the result¬
ing limit surface. This process of Gaussian noise addition is similar to that of the
successive random addition method mentioned earlier, the difference being the vor¬
tex positions obtained using butterfly subdivision rules are perturbed instead of grid
points obtained through midpoint interpolation.
Two natural terrain synthesis results using dynamic.1 modified butterfly subdivi¬
sion surface model are presented in this section. The initialized model is deformed
by applying spring forces on its limit surface from the discrete data points. At each
time stop, every control vortex position is perturbed by adding a random noise drawn
from a Gaussian distribution. The variance of the Gaussian distribution determines
the roughness of the synthesized surface. In both the examples, the initialized but¬
terfly subdivision model has an initial (control) mesh with 98 triangular faces and
68 control vortices. The “natural” looking limit surface of the initialized model is

154
Figuro 8.11. Synthesized natural terrain of different roughness using the dynamic
butterfly subdivision surface model with 841 vertices from a data set of 3099 elevation
values.
deformed by the forces exerted from the discrete elevation data points. When an
approximate fit is obtained, the model is subdivided to obtain a closer fit using more
degrees of freedom (control vertices) of the now initial mesh. The fitted surface has
1568 triangular faces and 841 control vertices in both the experiments. It may be
noted that synthesis of same quality natural terrains using the existing techniques
requires a largo number of grid points (of the order of 10°) [1. 82, 94, 103] and hence
the proposed technique provides a more compact representation of the synthesized
terrain. The elevation data values are sealed to fit an unit cube and the variance
of added noise is 10-4 for the synthesized fractal surface depicted in Fig.8.10. The
corresponding value of noise variance for fractal surfaces depicted in Fig.8.11(a), (b)
and (c) aro 10-6, 10-° and 10-4 respectively. In the first experiment, 4096 elevation
data points are used whereas the second data set comprised of 3099 elevation values.
The error in fit is approximately 1% in both the examples.

CHAPTER 9
CONCLUSIONS AND FUTURE WORK
In this chapter, the contributions of this dissertation are summarized. Future
research directions of the deformable subdivision surface model are also discussed.
9.1 Conclusions
In this dissertation, a dynamic framework has been developed for subdivision
surfaces. This dynamic framework enhances the applicability of subdivision surfaces
in geometric modeling applications where the modeler can directly and intuitively
manipulate the smooth limit surface using synthesized forces. The proposed frame¬
work is very useful for hierarchical shape recovery from large range and volume data
sets, as well as for non-rigid motion tracking from a temporal sequence of data sets.
Multiresolution representation of the initial mesh controlling the smooth limit sur¬
face enables global and local editing of the shape as desired by the modeler. A novel
technique for synthesizing natural terrain models from sparse elevation data using the
dynamic framework in conjunction with a variant of the successive random addition
technique has also been presented in this dissertation.
Several theoretical contributions has also been made in this dissertation. Local
parameterization techniques suitable for embedding the geometric subdivision surface
model in a physics-based modeling framework has been developed. Specific local
155

156
parameterization techniques have been fully worked out for Catmull-Clark. modified
butterfly and Loop subdivision schemes. Techniques for assigning material properties
to geometric subdivision surfaces have been derived, motion equation for the dynamic
model has been formulated, advantages of deformable models are incorporated into
conventional subdivision schemes, dynamic hierarchical control has been introduced,
multiresolution representation of the control mesh is derived and an unified approach
for deriving subdivision surface-based finite elements has been presented.
The proposed dynamic framework has a promising future in computer graph¬
ic's, geometric.1 modeling and scientific.' visualization. Furthermore, the finite element
techniques proposed in this dissertation should be of great interest to the engineering
design and analysis community as well.
9.2 Future Directions
The proposed dynamic.' framework enhances the applicability of the subdivision
surfaces in various applications, but still there are lots of research need to be done
to meet the growing demands of modeling and other applications. In the rest of this
section, several key issues needing attention are discussed.
9.2.1 Automatic Change of Topology
In the current framework, the evolving model can not change its topology if
needed. Automatic.' change of topology for dynamic.' subdivision surface models is
a very challenging and important research topic.'. The topologically adaptable de¬
formable models developed by Malladi et al. [54] and Melnerney and Terzopoulos

157
[63] can provide some insight about how to deal with topology (•hanging in deformable
subdivision surface models.
9.2.2 Local Refinement
In some applications, especially in shape recovery, it is more meaningful to
locally refine the smooth limit surface, i.e., introduction of new degrees of freedom
only in the regions where more details need to be recovered. Currently, this refinement
is global where new degrees of freedom is introduced throughout the smooth limit
surface. Local refinement implies distribution of degrees of freedom at a number of
subdivision levels instead of one particular subdivision level. This is a very challenging
but important research issue.
9.2.3 Automatic Model Parameter Selection
The model parameters like mass, damping, rigidity, bending are selected by the
user in the current implementation. The choice of force constants and time step
is also manual. Numerical stability of the deforming model is very sensitive to the
choice of the parameter values. Research needs to be done on how to select these
parameters automatically in different applications.
9.2.4 Constraint Imposition
In some applications, there might be a need to impose geometric.1 as well func ¬
tional constraints on the dynamic.1 subdivision surface models. Position and normal
constraints are examples of possible constraints. It is somewhat easier to impose these

158
constraints on interpolatory dynamic.1 subdivision surface models, but considerable re¬
search efforts need to be put for constraint imposition in the (.'ase of approximating
dynamic.1 subdivision surface models.
9.2.5 Recovery of Sharp Features
Recovery of sharp features using dynamic.1 subdivision surface models is an open
research problem. Even though piecewise smooth subdivision surfaces [40] has been
developed, incorporating sharp features in a physically meaningful way for an evolving
subdivision surface model is a challenging research issue.
9.2.6 Automatic Model Initialization
Currently model initialization is interactive. However, automatic.1 model initial¬
ization techniques have been recently proposed by Delingette [24], Similar techniques
for deformable subdivision surface initialization need to be developed.
9.2.7 Improved Synthesized Force Fields
Recently Xu and Prince [106] have proposed new type of force fields which makes
the model insensitive to the initialization. It would be nice to incorporate these force
fields in the current system for better performance in the model fitting context.

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BIOGRAPHICAL SKETCH
Chhandomay Mandal was born on 16th December. 1972, at Bishnupur, WB. In¬
dia. He received Bachelor of Technology (Honors) degree in electronics and electrical
communications engineering from Indian Institute of Technology. Kharagpur, India,
in June, 1995, and Master of Science degree in computer and information science and
engineering from University of Florida, Gainesville, in May, 1997. He is expected to
receive the Doctor of Philosophy degree in computer and information science and engi¬
neering from University of Florida, Gainesville, in December, 1998. He has published
several papers in technical conferences and reputable journals, and has served as an
auxiliary reviewer for IEEE Transactions on Pattern Recognition and Machine In¬
telligence and for several conferences, including International Conference on Pattern
Recognition and IEEE Conference on Computer Vision and Pattern Recognition. His
research interests include computer graphic's, computer vision, physics-based model¬
ing and simulation, computer-aided geometric.1 design, visualization techniques and
computer animation. He is a member of ACM SIGGRAPH.
168




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*&*%# &! +(#" +.-%*+ %$ #% #)! +*+ % !%&' "%))*&!* #"* (.#* !%" +, (.#*2% '*' !*(*# '*+ !& %&' + *& # > / +.%) 6%# & *+*% "*+0 "* *4 +# &! #* "& A.*+ ('*) &! +(#" ()*4 +"%*+ %& $* $%') )%++ >*' & # # 1 ' +# &# %#*! *+ &%(*) , BC *4) # %# " &! .+ &! %%(*# +.-%*+ %&'B7C +.$' / + &+.-%*+0 "* *4) # %# " &! #* "& A.* & / )/ *+ %# # & &! #"* ('*) +.-%* & # % ))*# & &' / '.%) %%(*# +.-%* %# "*+0 '5%*& # +.-%* %# "*+ %* #"*& *4) #) +# # "*' #!*#"* .+ &! & # & . # &+#% & #+0" + *4) #+# # " &! *++ + / * () %#*' ('*) &! +(#" +.-%*+ %$ #% #)! '.* # #"* & # & . # &+#% & #+ 1" " &**' #$* +%# +>*' %)&!#"* %# " $.&'% *+0 .$' / + &+.-%*+%*+ ()**'.%)('*)+1" "?*%&%)#*&%#**2 *+*& #%# & #"* +(#" +.-%*+ %$ #%#)! 0 # %)*.+ / * +.$' 2 / + & + "*(* '.*+ % / +.%)) )*%+ &! +(#" +.-%* & #"* ) ( # $ **%#*' %) %# &% >4*' +*# *>&*(*& # .)*+ & %& .+*2'*>&*' & # %)& #) (*+"0 "* & # %) & #) (*+" + % + ()* )!&%) (*+" #"* +%(* #)! %) # * %+ #"%# #"* +(#" +.-%* # $*('*)*'0 # *% " +#* +.$' / + &, % )!&%)(*+"1 #"(*/ *# *+%&'-%*+ +&+#.#*'-(#"**/ .+&*/ %

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7 #"* *>&*(*& # *++, %&' #"* +(#" +.-%* + $#% &*' & #"* ) ( #0 1 */ *, % -*1+.$' / + &+#*+&#"* & # %)(*+"!*&*%))+.Æ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Æ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Æ2 *& # +).# & # #"* +"%* * / * $)*( %+ 1 *))0 + ()* +.$' / + & +.-%*

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Æ *& #) * / *+ +"%*+%+ 1 *)) %+ &&2 ! '(# &+-( )%!*%&!* %&' / ).(*# '%#%+*#+0 #* #"%##" + ' ++*#%# &

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< $#" #"* " + +2$%+*' ('*) &! " )+" %&' #"* !*(*# +.$' / + & + "*(*+ %* &%#*'1 #" &% + &!)*.& >*' -%(*1 30 +*+1 ))$*%$)*#(%& .)%#*#" +" + +2$%+*'('*) &%&%$ #%*! &, %&' #"* ('*) 1 )) *+&' &%#.%)) B5.+# ) 3 * % *%)21 )' $5*# 1 .)'C # #" + -* %) %# &0 " + +"%* '*-(%# & + A.%& # #%# / *) "%%#* 6*' $ #"* # (*2/ % &! ' +)%*(*& # & #) & #+ #"%# .& A.*) '*>&* #"* !*(*# -#"* ) ( #+.-%*0 "* '&%( -%(*1 3 1 % / *)*#+ '* / *' .+ &! +.$' / + & + "*(*+ *&2 %$)*+ %( .)# *+).# &**+*& #%# & -#"* & #) (*+" '*>& &!#"* +(#" ) ( #+.-%*0 " + / '*+%'' # &%)D*4 $ ) # ##"*" + +2$%+*'('*) &!-%(*1 3 + &* #"* ('*)* + -** # "+* #"* '*+ *' *' # &! )*/ *)0 % ) 1 **+).# &*' # &!)*/ *) + "+*&,#"*+& #"*+ 6*'-*%) %# &&#"* +(#" ) ( #+.-%* 1 )) "% / * !)$%) *?*#+, 1"**%+ *' # &! 1 #" -*2$%+*' #)+%#%>&* *+).# & 1 )) *)'%)%) 6*'*?*#0 )! #"(+%&'*'.*+%*'*/ *)*'1" "% 4 (%#*#"*+*'&*1 >& #**)*(*& #+.+ &!%))*# &-) &*%%&'I$ ) &*%>& #**)*(*& #++.$5*# # #"* () # !*(*# &+#% & #+ *&-*' $ #"* +.$' / + & .)*+0 " + " *% " %))2+#.#.*'% 4 (%# &%&+%# +-%& .+*2+* >*'*#)2 *%&*0 "*+*''&%( -%(*1 3*&"%&*+#"*%) %$ ) # -+.$' / + &+.2 -%* ('*)+ & / % .+ %) %# & %*%+0 # / '*+ % ' *# %&' & #. # / * 1 % -

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(%& .)%# &! +"%*+ & % " *% " %) -%+" & !*(*# ('*) &! %) %# &+0 # "%+ %)+ $**& +.*++-.)) .+*' *Æ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8 "* +*' ('*) &!-%(*1 3 + .+*' & / % .+%) %# &%*%+ %&' #"* *+.)#+ %* *+*& #*' & "%#* 0 "* +*' '&%( ('*) "%+ $**& +.*++2 -.)) .+*' & !*(*# ('*) &!, +"%* * / * %&' &&2 ! ' (# & *+# (%# & %) %# &+0 & / *) #* "& A.* &%#.%)#*% &('*) &! + %)+*+*& #*' 1"** #"* '&%( -%(*1 3 + ( $ &*' 1 #" % / % %& # % -%#%) +.-%* +& #"*+ + #* "& A.*0 &%)) ,&).+ &+ %* '% 1& & "%#* 1"** -.#.* ' *# &+ -*+*% " %* %)+ & #*'.#0

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7 &#" + "%#*,%$ */ */ *1-#"*+.$' / + &+.-%*+ +*+*& #*'-)) 1 *' $ % */ *1 #"* */ .+ 1 3 '&* & #"* %*% +.$' / + & +.-%*+0 " + + -)) 1 *' $ %& / */ *1 #"* " + +2$%+*' ('*)+ %)&! 1 #" % */ *1 #"* *2 )%#*') #*%#.*0 "*(# / %# &!-%#+-*( $*'' &!!*(*# +.$' / + &+.-%* ('*)+ &%" + +2$%+*'('*) &!%%' !(%**+*& #*' &#"*&*4#+*# &0 "* %'/ %& #%!*+ #"* +*' '&%( ('*)+ / * #"* *4 +# &! " + +2$%+*' ('*)+ +"%* * / * %* ' +.++*' %&' >&%)) %'/ %& #%!*+ % .& >*' -%(*1 3 +"%* ('*) &!%&' +"%* * / * %*%)+*+*& #*'0 70 .$' / + &.-%*+ "* &.# # %& +.$' / + & + "*(* + %& & # %)(*+" B%)+ 3& 1& %+ & #) (*+"C, O B C, 1" " + % ))*# & / *# *+ %&' % ))*# & -%*+ 0 "* +.$' / + & +.-%* B C %++ %#*' 1 #" #"* & # %) (*+" + '*>&*' %+ #"* ) ( # #"* *.+ / * %) %# & #"* *>&*(*& # %+ +" 1& & !0700 "* *>&*(*& # , 1"*& %) *' & % (*+" O B C, *%#*+ %*>&*' (*+" OB C1"**#"*/ *# *+ & %*(.#*'%+%Æ&*( $ &%# &+ #"* / *# *+ & %&' #"* -%*+ & %* $#% &*' $ +) ## &!*% " -%* & & #% >4*' & .( $* -+.$2-%*+0

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

PAGE 45

=8 %$ #%#)! ,#"*+*+.$' / + &+.-%*+(% &#$**%+ )%%(*#* 6%$)*%&' '* / &!%)+*'2-(%&%)# -( .)%# &-#"* ) ( #+.-%* %& $*/ * ' Æ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Æ&* ( $ &%# &-& #)/ *#*4 + # &+ &#"* & # %)(*+"0 "**-*, #"* '&%( $*"% / #"* ) ( #+.-%* %& $* & #))*' $ -( .)%#2 &! #"* '&%( ('*) & #"* & # %)(*+" #+*)-, #"* &) *4*# & $* &! #"* %+* 1"*& #"* & # %)(*+" "%+ &&2*#%&!.)%-%*+0 " + $)*(%& $* .(/ *& #*' $ #%3 &! #"* (*+" $#% &*' #".!" &* +#* +.$' / + & %+ #"* & # %) (*+"0 # (% $* &#*' #"%# %& %'' # &%) +.$' / + & +#* (% $* &**'*' & +(* %+*+

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

PAGE 55

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

PAGE 56

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:7 *# %& $* 1 ##*&%+ OB CB C B=07C 1"** O B C%&' O 0 "* '*%-)%)*.+ / * +.$' / + & %.&' %& *4#%' &% & # + )).+#%#*' & !0=0;0 #* #"%#, *% " / *#*4 + # & &#"*+.$' / '*'(*+" +$#% &*'$ %&%Æ&*( $ &%# &-+(* / *# *+ & #"* */ .+ )*/ *) %&' "*&* %& 1 +.(+ # 0 "* )%!*+# * !*& / %).* +. " % (%# 4 + %&' # %& $* +" 1& #"%# #"* *+&' &! &>& #* +* *+ + & / *!*& # -)) 1 &!% + ( )%%% " %+ & %)+#*%' *# %)0 M=N0 "* *+# #"* '* / %# & )*%' &! # %& *4*++ & + *)%# / *) +#% !" # -1 %'0 + &! #"* +%(* %% " .+*' # '* / * #"* A&0=0<, #%& $* +" 1& #"%# O B CB COB B CB CC OB C B=0=C (A&0=09, A&0=0< %&'A&0=0=, OB C GB C B=0:C *# OB CGB C, %&' "*&* O B=0;C

PAGE 58

:= spj}0{spj}0{spj}0{1 jC}s{1 jC}s { {jBs 1} {jBs 2} 2{jBs 3} !.*=0;0 %)+.$' / + &%.&'#"**4#%' &% & #%&'#"**+&' &! %# "*+ &#"* ) ( #+.-%* -(' ?**& # )*/ *)+ -+.$' / + &0

PAGE 59

:: =070: &%( + & #"* */ .+ +*# &, %& *4*++ & + '* / *' #"* +(#" ) ( # +.-%* $#% &*'/ % &>& #*+.$' / + &+#*+0 " +*4*++ &#*%#+#"*+(#") ( #+.-%* %+ % -.&# & #"* & #) / *#*4 + # &+ & #"* & # %) (*+"0 1 #"* / *#*4 + # &+ & #"* & # %) (*+" '*>& &! #"* +(#" ) ( # +.-%* %* #*%#*' %+ % -.&# &-# (* &'*#*( $*'#"*%#( .))2)%3+.$' / + &('*) &%'&%( -%(*1 30 "* / *) # -#" ++.-%* ('*)%&$* *4*++*' %+ S B CO S B=09C 1"** %& / *+#. 3 '# '*&#*+ % # (* '* / %# / *0 "* " + + #"* '&%( +.$' / + &+.-%*('*) +$%+*'&#"*1 32*&*!/ *+ &-%!%&! %&'&%( + M=9N %&' + -( .)%#*' &%& %&%)!.+ 1 % # #"* '&%( -%(*1 3 *+*& #*' & *# & 7070 *# B C$*#"*(%++'*&+ # -.&# &-#"*+.-%*0 "*&#"*3 &*# *&*! #"* +.-%* + O 7 S S O 7 S S B=0
PAGE 60

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

PAGE 61

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

PAGE 62

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

PAGE 63

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

PAGE 64

: =0= & #*)*(*& # ()*(*& #%# & "**/ ).# &-#"*!*&*%) 6*'' &%#*+-.&*1'&%( +.-%*('*) %&$*'*#*( &*'$ #"*+*&'2'*' ?**& # %)*A.%# &%+! / *&$ A&070;0 & %&%)# %)+).# &-#"*! / *& &!' ?**& # %)*A.%# &%&&#$*'* / *' &!*&*%)0 1 */ *, %&*Æ *& # & .(* %) ()*(*& #%# &%&$* $#% &*'.+ &!#"* >& #**)*2 (*& #(*#"'M:7 N0 "*) ( #+.-%*-#"*'&%( %#( .))2)%3+.$' / + &('*) +%))*# &-$ .$ %# "*)*(*& #+0 *.+*# 1 # *+ ->& #**)*(*& #+-#" + .+*2&(%)*)*(*& #+B$ .$ %# "*+%++ !&*'##"*&(%)-%*+-#"* & # %) (*+"C%&'+* %)*)*(*& #+B))*# &&>& #*& .( $*-$ .$ %# "*+%++ !&*' # *% " *4#%' &% / *#*4 #"* & # %) (*+"C0 "* +"%* -.&# &+ #"* &2 (%) *)*(*& # %* #"* $%+ + -.&# &+ #"* $ .$ +.-%* %# ", 1"**%+ #"* +"%* -.&# &+ #"* +* %) *)*(*& # %* #"* ))*# & $%+ + -.&# &+ *+&' &! # #"* $ .$ %# "*+ & #"* +* %) *)*(*� & #"* .*& # ()*(*& #%# &, #"* , %&' (%# *+-*% " &' / '.%)&(%)%&'+* %)*)*(*& #+%*%).)%#*' %&'#"* %&$*%++*( $)*' & ##"*!)$%) , %&' (%# *+#"%#%*% &#"* *+&' &!' +*#**A.%# &-(# &0 &%# *,1 *&*/ *%++*( $)*#"*!)$%) (%# *+*4) #) &#"* & #**+#-# (**-(%&*0 "*'*#% )*' ()*(*& #%# & + *4)% &*' &#"* -)) 1 &!+*# &+0 =0=0 %#%#.#.*+ "*) ( #+.-%*-#"*'&%( %#( .))2)%3+.$' / + &('*) +%))*# & -$ .$ %# "*+,%&'"*&**A. *+.+#3 **#% 3-#"*& #))!&+'*>& &! +. " %# "*+0 * %& !*# #"* & #) )!&+ #"* &(%) *)*(*& #+ & #"* ) ( #

PAGE 65

;8 list of faces list of edges list of normal elements list of vertices list of faces list of edges list of normal elements list of vertices list of faces list of edges list of normal elements list of vertices Subdiv object Subdiv object local subdivision around extraordinary vertices Subdiv objectNULL !.*=090 "*'%#%+#.#.*.+*'-'&%( +.$' / + &+.-%* ()*(*& #%# &0 +.-%*-(#"* & # %)& #)(*+" #+*)-0 1 */ *,1 *&**'#)%))+.$' / '*#"* & # %)& #)(*+"%.&'#"**4#%' &%/ *# *+#$#% &#"*& #))!&+ #"* $ .$ %# "*+ &#"* +* %) *)*(*& #+ &#"* ) ( #+.-%*0 +.$' / + & +.-%* '*>&*' $ % & #) (*+" %# %& )*/ *) + '*+ !&*' %+ % )%++1" ""%+ % & #*# #+%*& #(*+",%+*# & #*+# #+?+ &!(*+"*+ B% + &!.#-)%)+.$' / + &%.&'#"**4#%' &%/ *# *+%##"%#)*/ *)C,%) +# --%*+,*'!*+,/ *# *+%&'&(%)*)*(*& #+0 %*,*'!*,/ *#*4%&'&(%)*)*(*& #+ %*, &#.&,)%++*+1" "+#*%))#"*&&*# / # %&'#"* &-(%# &&**'*'# * #"**& .(*%#*%))#"*%# "*+)%))+.$' / '*%.&'%&*4#%' &%/ *#*4 & #"%# )*/ *)0 "* ()*(*& #%# & #%3 *+ #"* & # %) (*+" %+ #"* $%+* +.$' / + & +.-%*$5*#B1 #" #+%*& # & #*+*##C%&')%))+.$' / '*+#"* & # %) (*+".#%.+*2'*>&*'(%4 ( .()*/ *)%.&'*% "*4#%' &%/ *#*4#*%#*

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� =0=0= * %))*(*& #+ % " +* %) *)*(*& # &+ +#+ %& &>& #* & .( $* $ .$ %# "*+ & #"* ) ( #0 * "% / * %)*%' '*+ $*' % *.+ / * *& .(*%# & #"* $ .$ %# "*+ %+* %)*)*(*& # &*# &=07070 *#.+& 1&+ '*%&%$ #%$ .$ %# "#"*+* %)*)*(*& # &+(*)*/ *) 0 "*(%++(%# 4 -#" +%# "%&$*1 ##*& %+ O B=07C 1"** + #"* &(%) *)*(*& # (%++ (%# 4 B+%)*' $ % -%# # #%3 * & # %.& # #"* %*% +" &3 %!* & $ .$ %# "*+ %# " !"* )*/ *) +.$' / + &C %&'

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;= %) *'+ &! -* $* &! B CO B B CC Æ B C B=07C 1"** Æ +#"*.& # (.)+*-.&# & () &! B CO B C %&'/ %& +"*+ *)+*1"** & #"* +.-%*0 1 */ *, #"* Æ -.&# & %& $* *)%*' 1 #" % +(#" 3 *&*)#+*%'#"*-* / *%!*%#*# &&#"*+.-%*0 "*+ &!-*+%& $* %) *' & #*%# / *) .+ &! #"* (.#* (.+* #"* & #+ -( 1" " -*+ &**' #$* %) *' %&$**%' &-(%>)*0 * / *+"%*+-(= (%!*'%#%,1 *+& #"*+ 6* (%!*2$%+*'-*+0 = *'!* '*#*# & +*-(*'&%/ ).(*'%#%+*# .+ &!#"*=&!%2* "*BC *%# M9N # '.* % =#*& # %) >*)' B C, 1" " 1 * .+* %+ %& *4#*&%) #*& # %)-#"* ('*)0 "* -* ' +# $.# & +#"*& (.#*' %+ B CO & B C B C B=0=8C 1"** & & #)+#"*+#*&!#"-#"*-*0 "*%) *'-*&*% "/ *#*4%##"* 2 #"% 4 (%# &)*/ *) +(.#*'$ # ) &*% & #*)%# &-*/ %).%# &!A&0=07= & %#*+ %& ' &%#*+0 * +" +# %#*' (%!*2$%+*' -*+ 1" " &%#* *! &2$%+*' &-(%# &+. "%+!%' *& #+-%#"*+")'*'-.66/ 4*))%++ >%# & %& %)+$* .+*' # *)'$*##* %&' (* %.%#* +"%* * / * 0 # (% $* &#*' #"%# 1 * %& %) + &! -*+ & %'' # & 1 #" #"* (%!*2$%+*' -*+ $ )% &!

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;: & #+&#"*$.&'%-#"**! && #**+# &*% "+) *+-#"*=/ ).(*B, *#0C (%!*'%#%0 =0=0; +*#*&%( A.%# & "* ' ?**& # %) *A.%# & ! / *& $ A&070; + & #*!%#*' #".!" # (* $ ' +2 *# 6 &! #"* # (* '* / %# / * / * # (* +#*+ T 0 "* +#%#* #"* '&%( +.$' / + & +.-%* %# # (* GT + & #*!%#*' .+ &! +#%#*+ %# # (* %&' T 0 & () # # (* & #*!%# & (*#"' + .+*' & #"* .*& # ()*(*& #%# & 1"** ' +*#* '* / %# / *+%* %).)%#*'.+ &! Q B GT CO B GT C 7 B CG B T C T B=0=C %&' S B GT CO B GT C B T C 7T B=0=7C "* *)*(*& #%) (%++, '%( &! %&' +# ?&*++ (%# *+ %& $* %++*( $)*' # !*# #"* !)$%)(%++,'%( &!%&'+# ?&*++(%# 4-#"*+(#"+.$' / + &+.-%*('*)0 1 */ *, 1 * ' &# %++*( $)* #"*+* !)$%) +%+* (%# *+ *4) #) *Æ *& *%+&+0 #"* # (* / % &! +# ?&*++ (%# 4, 1 * *(.#* %# *% " # (* +#*0 + &!A&070;,A&0=0=%&'A&0=0=7,#"*' +*#**A.%# &-(# & +$#% &*'%+ B7 G T G7T C B GT CO7T B GT CGB T 7 C B T CG: B C B=0==C

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;; with more patches same limit surface evolution(a) (b)evolution evolution subdivision at equilibrium evolution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Æ *& #) +(%))%&' #"* "%&!* & ('*)*&*! $*# 1 **& +.*++ / * #*%# &+$*(*+ )*++ #"%& %*2+* >*' #)*%&*0 =0: *&*%) 6%# &-#"* % " "*+*' %% "%&$*!*&*%) 6*'-#"*% 4 (%# &!+.$' / + & + "*(*+0 1 */ *,%(*!*&*%)%% " +*+*& #*' &"%#*;-'* / &!#"* '&%( -%(*1 30 &%( %#( .))2)%3 +.$' / + & +.-%* ('*) + *-( .2 )%#*' & *# & ;07 .+ &! #" + !*&*%) %% "0 '&%( -%(*1 3 %&#"* .)%% 4 (%# &!+.$' / + &+ "*(*&%(*) ,H++.$' / + &+ "*(*, +%)+ ' +.++*' &"%#* ;0

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; !.* :00 B%C "* & #) )!& 1 #" # %&!.)% -%*+F B$C (*+" $#% &*' %-#* &* +.$' / + & +#* .+ &! $.##*D +.$' / + & .)*+0 12 2 3 4 8 7 6 5 -w 0.5 0.5 2w 2w (a) (b) -w -w -w j+1 v jvjvjvjejvjvjvjv1 !.* :070 B%C "* & # $.# &! / *# *+ & #"* 52#" )*/ *) #"* / *#*4 & #"* 5G2 #" )*/ *) *+&' &! # #"* *'!* $*# 1 **& %&' F B$C #"* 1 * !" &! -%#+ ' ?**& # / *# *+0

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; + + )# $ %'' &! % &*1 / *#*4 1"+* + # & + $#% &*' $ %& %Æ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98 w w 0.125+2w 0.125+2w -0.0625-w (a) (b) q-0.0625-w -0.0625-w -0.0625-w 0.5-w 0.5-w n-1s n-2s1s0s2s4s3s !.* :0=0 B%C "* 1 * !" &! -%#+ & # $.# &! / *#*4 + # &+ %& *'!* &&*# &!# 1 / *# *+-'*!**9FB$C#"**+&' &!%+*1"*&&*/ *#*4 +'*!** %&' #"* #"* +-'*!** 90

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97 x x x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; $.##*D +.$' / + &('*)*+*& #*' &"%#*: +#%3 *&()*(*& #%# &.2 +*+0 & % 4 (%# & #"* +(#" ) ( # +.-%* + $#% &*' $ +.$' / ' &! #"* & # %)& #) (*+" # (*+, %&' % + &!2(%++ ++#*( + '*/ *)*' & #" + 2#" % 4 (%# & )*/ *) & % + ( )% -%+" & %+ & *# & :0=0 "* " + %) (%# *+ #" + ++#*( %* .+*' %+ % 4 (%# &+ # #"* %#.%) " + %) (%# *+0 "*+* % 4 (%# &+"% / * $**& -.&'#$* / * *Æ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9 !.* ;0:0 B%C "* & #) )!& 1 #" # %&!.)% -%*+F B$C (*+" $#% &*' %-#* &* +.$' / + & +#* .+ &! H++.$' / + &.)*+0 1"** , , %&' %*#"*+ # &/ *#-#"*/ *#*4 , , %&' *+*# / *) 0 *1 *'!*+ %* -(*' $ &&*# &! *% " &*1 / *#*4 & # # #"* &*1 *'!* & #+*+&' &!##"**'!*+ & '*& #&#"*)'/ *#*4,%&'$ &&*# &! *% "&*1 *'!* & ###"*&*1 *'!* & #+-#"*#"**'!*+ &#"*# 1 -%*+ 1" " +"%*' #"* ! &%)*'!*0 *1 -%*+ %* '*>&*' %+-%*+ *&)+*' $ #"* &*1*'!*+0 %()*+ *>& &! %& & # %) (*+" .+ &! H+ +.$' / + & .)*+ %* +" 1& & !0;0: %&' ;0;0 "*+* +.$' / + & .)*+ *&+.* #%&!*& # )%&* & # & . # #"* ) ( # +.-%* */ *& & % *!.)% +*## &!, 0*0, 1"*& #"* # %&!.)% & #) (*+" "%+

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control mesh limit surface !.*;0;0 & & # %)(*+"%&'#"**+&' &!) ( #+.-%*$#% &*'.+ &!H+ +.$' / + &.)*+0 "* '(% &+-#"*+"%'*' # %&!.)%%# "*+ &#"* ) ( #+.-%* %* #"**+&' &! # %&!.)%-%*+ &#"* & # %)(*+"0 "* *& )*'/ *# *+ %* #"* & #)/ *# *+ -#"* *+&' &! # %&!.)%%# " &#"* ) ( #+.-%*0

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limit surface initial mesh(1,0) (0,1) (0,0) !.*;090 % " # %&!.)%%# " & #"* ) ( #+.-%* %& $* %++ %#*' 1 #" %&&2 $.&'%# %&!.)%-%* &#"* & # %)(*+",1" " &#.&%&$*%%(*#* 6*' / * % # %&!)*1 #" / *# *+ %# B8 8C,B 8C%&' B8 C0

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8 % 4 (%# &0 1 */ *, % %# %) %)#*&%# / * ()*(*& #%# & + # +.$' / '* #"* & #) (*+" # (*+ .+ &! H+ +.$' / + & .)*+, %&' # $. )' % + &!2(%++ ++#*( & #" + 2#" )*/ *) % 4 (%# & %+ "%+ $**& '&* #"* '&%( (' >*' $.##*D+.$' / + &('*) &*# &:0=0 "*" + %)(%# *+-#" ++ &!2(%++ ++#*( / '*%&% 4 (%# &-#"* ! &%)" + %)(%# *+,%&' #1 3+1 *)) &%# *0 ;0: *&*%).#) &*-#"* %(*1 34 (%# &!.$' / + & "*(*+ "* .& >*' %% " *+*& #*' # / '* % '&%( -%(*1 3 %#( .))2 )%3%&'+.$' / + &+ "*(*+%&$*!*&*%) 6*'*%+ ) 0 " +%% " & / )/ *+ #"** +#*+ 1" " %*%+ -)) 1+E 0 "* ) ( # +.-%* $#% &*' / % %& % 4 (%# &! +.$' / + & + "*(* %& $* *4*++*' %+%))*# &-+(#"%# "*+1" "%&$*)%))%%(*#* 6*' / * % *+&' &! -%* & #"* & #) (*+"0 % " %# " + 2+ '*' # + )%)) %%(*#* 6*' / * % 2+ '*' -%*0 &%)# *4*++ &+ *% " #"*+* %# "*+ %& $* '* / *' */ *& & #"* *+*&* *4#%' &% / *# *+ & #"*& #)(*+",%&'"*&*%&*4*++ &-#"*) ( #+.-%*%&$*$#% &*'0 70 &*%&*4*++ &-#"*) ( #+.-%* +$#% &*',#"*'&%( -%(*1 3%& $* '*/ *)*' $ #*%# &! & #) / *#*4 + # &+ %+ % -.&# & # (*0 "* *+&' &! (# &*A.%# & %& $*'* / *'0

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Æ *& *%+&+0

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9 "* '&%( -%(*1 3 '*/ *)*' +.$' / + & +.-%*+ %)) 1+ #"* .+*+ # ' *#) (%& .)%#*#"* +(#") ( #+.-%* $ %) &! +& #"*+ 6*' -*+0 & %& +.$' / + & + "*(*, %& & # %) B& #)C (*+" + *>&*' .+ &! +* > +.$' / + & .)*+ # $#% & % +(#" +.-%* & #"* ) ( #0 '*/ *) #"* '&%( -%(*1 3, #" ++(#") ( #+.-%* +%%(*#* 6*' / *#"*'(% &'*>&*'$ #"* & # %)(*+" %&' "*&*, #"* ) ( #+.-%* %& $* *4*++*' %+ % -.&# & #"* & # %)(*+"0 "* .+*+ %) +& #"*+ 6*' -*+ & #"* +(#") ( #+.-%* 1" " + #%&+-**' $% 3 &#"* & # %)(*+"/ %%#%&+-(%# &(%# 40 "* & # %)(*+"*/ )/ *+ / *# (* & *+&+*##"*%) *'-*+,%&'&+*A.*&#)#"*+(#") ( #+.-%*'*-(+# $#% & %& *A. ) $ .(+ # & 1"** %)) -*+ %* $%)%&*'0 + &#*' *%) *, */ *& #".!"/ % .+# *+ --*+%&$*%) *'&#"*) ( #+.-%*#$#% &%'*+ *' *?*#, #"* ++ $)* +"%*+ #"%# %& $* $#% &*' / % */ ).# & + ' *#) *)%#*' # #"* & .( $* & #)/ *# *+ *+*& # &#"* & # %)(*+"0 "*&*#-( .)# )*/ *)'&%( +1 %+ & #'.*'*%) * &'*#"% / */ %2 &! & .( $* & #) / *# *+ & #"* & # %)(*+", %&' % 1 '* %&!* +"%*+ 1 #" %$ #% #)! %& $* ('*)*' 1 #" & #"* '&%( -%(*1 30 "* *%'* + 8=

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8: synthesized by the user force applied on the smooth limit surface force is applied back to the transferred initial (control) meshposition equilibrium to obtain evolves mesh the initial equilibrium surface at limit smooth deformed shape detailed a more to model user needs previous initial mesh is replaced by a new initial mesh with more degrees of freedom obtained via subdivision on the previous initial meshblack box to the user !.*900 "*(%# $) 3 ' %!%(-#"* ( .)# )*/ *)'&%( + %% "0

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8; *-**'#*# &=070;%&'*# &=0=09-#" +' ++*#%# &-#"*'*#% )+-( .)# 2 )*/ *)'&%( + &#"*& #*4#-'&%( %#( .))2)%3+.-%*+,%&'#*# &:070= -#"*'&%( (' >*'$.##*D+.$' / + &+ "*(*0 "*&*#-( .)# )*/ *)'2 &%( + + )).+#%#*' 1 #" $) 3 ' %!%(+ & !0900 " + ( .)# )*/ *) '&%( + %& $* &+ '**' %+ % +.$' / + & +.-%*2$%+*' %+*2#2>&* ('*) &!-%(*1 3 %&' %&$*+.((% 6*'%+-)) 1+0 "*.+*+#%#+1 #"%+(#") ( #+.-%*1" ""%+ %/ *+ ()* & # %)(*+"1 #"-*1& #)/ *# *+0 " ++(#"+.-%* +'*-(*' $ %) &! +& #"*+ 6*' -*+ + #"%# #"* ) ( # +.-%* %# *A. ) $ .( 1 .)' )3 ) 3 *#"*>&%)'*+ *'+"%*$.#1 #".##"*'*#% )+ & #0 "*&,#"*& .( $*-& #) / *# *+ & #"* & # %)(*+" + &*%+*' $ *)% &! #"* & # %)(*+" %# *A. ) $ .( $ %&#"* &* $#% &*' / % &* +.$' / + & +#* & #"* )' & # %) (*+", %&' #"* )' & # %) (*+" + ' +%'*'0 #" #"*+* )' %&' &*1 & # %) (*+" **+*& # #"* +%(* ) ( # +.-%*, $.# #"* )%##* "%+ (* & #) / *# *+0 1, #" + &*1 & # %) (*+",%&'&+*A.*& #)#"*+(#") ( #+.-%*,%&$*'*-(*'#$#% &%+"%* %# *A. ) $ .( 1" " "%+ (* '*#% )+0 " + *++ + **%#*' # )) #"* '*-(*' +(#"+.-%* "%+ %))#"* '*+ *' '*#% )+0 "*) ( #%# &-#"*( .)# )*/ *)'&%( +%% " +#"%#&*%>&**+).# & + "+*&,#"*('*)*%&&##-%) 1 **+).# &%+#"*'&%( + +'*>&*'& #"*.*& # & # %)(*+"0 "** +&1 % -"% / &!#"*'&%( +'*>&*'&%+ ()* & # %) (*+", %# #"* +%(* # (* *+*/ &! #"* %)*%' $#% &*' '*#% )+ & #"* ('*) 1 #" &#"*( .)# )*/ *)'&%( +%% "0 " + +#*& # %))%+*/ ***+# # &0 *4%()*,)*#.+ (%! &*#"* +*&% '* #*' & !09070 "* ('*)*"%+ * / **'

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89] [] [(static) (static) initial shape recovered global shape recovered detailed shape with few degrees of with large degrees of freedom freedom formulate dynamics on a coarser level of the control mesh with few degrees of shape, and then add back the details. freedom, bend the global (a) Shape recovery Multilevel dynamics approach :(time consuming, difficult)Multiresolution dynamics approach :(fast, easy)(b) Editing the recovered shape : Global bending + details + details formulate dynamics on the bend the entire detailed shape. current control mesh which has large degrees of freedom, !.*9070 .)# )*/ *)'&%( +/+0 ( .)# *+).# &'&%( +0

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8< % )&! +#% !" # *2) 3 * +"%* 1 #" )#+ '*#% ) $ +& #"*+ 6*' -* %) %# & & % +(#" ) ( # +.-%*0 " + +(#" ) ( # +.-%* & # %)) "%' % + ()* & # %) (*+" 1 #" -*1 / *# *+, $.# #"* .*& # & # %) (*+" %-#* (%& ('*) +.$' / + & +#*+ "%+ )%!* & .( $* & #) / *# *+0 1, #"* ('*)* 1 %& #+ # $*&' #" + +#% !" # *2) 3 * +"%*, #"* +& #"*+ 6*' -*+ %) *' & #"* +(#" ) ( # +.-%* &**'#'*-(%& & # %)(*+"1 #"%)%!*& .( $*-& #)/ *# *+0 #1 .)'"% / * $**& ( . " *%+ * %&' -%+#* % ) 1 * *+).# & / *+ & #"* .*& # & # %)(*+" .)' $* '*-(*' 1 #" #"* '*#% )+ $* &! %''*' $% 3 1"*& #"* +"%* "%+ $*& # # #"* '*+ *' *4#*& # & *+&+* # #"* %) *' +& #"*+ 6*' -*+0 )+, # "%+ $**& & #*'.# &*# &70;#"%##"*'&%( -%(*1 3%& & #*!%#*+"%** / * %&' +"%* ('*) &! &% .& >*' -%(*1 3 1"** #"* * / **' +"%* %& *%+ )$* *' #*'%+1 *))0 1 */ *, -#"*('*)*1 %& #+%!)$%) "%&!* &#"** / **'+"%*, % ) 1 * *+).# & & #) (*+" B#"%& #"* &* $#% &*' / % +"%* * / * *++C (% $* *A. *', %&'#"* '*#% )+&**' #$* *+*/ *' %##"* +%(*# (*0 ( .)# *+).# &**+*& #%# & -#"* & # %)(*+" ++*' & #" + "%#* #+)/ *#"*%$ / *2(*& # &*'$)*(0 #*A. ) $ .(,%& & # %)(*+" ++.$' / '*' # $#% & %& & # %) (*+" 1 #" (* & #) / *# *+ B'*!**+ -**'(C0 &+#*%' '*/ *) &! #"* '&%( + & #" + &*1) $#% &*' & # %) (*+" %+ & #"* ( .)# )*/ *) '&%( + %% ", #"* '&%( + + '*/ *)*' & % **+*& #%# & 1" " / *1+ #"* &*1) $#% &*' & # %) (*+" %+ #"* */ .+ & # %) (*+" ).+ '*#% ) B1 % / *)*#C 2 *Æ *& #+ &**'*' # !*# #"* .*& # & # %) (*+" B+** !090=C0 " + + *++*& # %)) '*/ *) &! #"* '&%( + & % ( .)# *+).# & **+*& #%# & #"* */ )/ &! & # %)

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8 ( 6 control vertices) ( 3 control vertices) degrees of freedom = 3 ( 3 control vertices, 3 detail coefficients) ( 3 control vertices) degrees of freedom = 3degrees of freedom = 6 degrees of freedom = 6(b) Multiresolution approach (a) Multilevel approach !.* 90=0 **+*& #%# & #"* '*!**+ -**'( & ( .)# )*/ *) '&%( + %&' ( .)# *+).# &'&%( +%% "0

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7= a bWavelet associated with the edge connecting aandbLazy : the scaling function associated with the introduced vertex at higher level Lifted : the scaling function associated with the introduced vertex at higher level ( weighted combination of the scaling functions associated with the solid vertices ) !.*9090 % / *)*# &+#.# &0 +%& '*& # # (%# 4& #*)%#+.$' / + &+ "*(*+,%&'"*&*$#"+& #"*+ + %&' %&%)+ + %& $* '&* & ) &*% # (* & #*)%# +.$' / + & + "*(*+0 "* + () >*' *4*++ &+ #"* +& #"*+ + %&' %&%)+ + >)#*+ & %+* & #*)%# +.$' / + &+ "*(* %*! / *& $ M N O ! ! B9077C %&' O ! ! B907=C *+*# / *) 0

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Æ *& #+0

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79 -.+*,#"*'*#% )*Æ *& #+%*6*1"*&%&& #)(*+" +*)%*'$ %&#"* & #)(*+"$#% &*'$ %.*+.$' / + &+#*&#"**/ .+(*+",$.#$*(*+ &&26*%+#"*&*1 & #)(*+"B**+*& #*' %+*/ .+& #)(*+"%&'1 % / *)*# *Æ *& #+C*/ )/ *+ / *# (* &*+&+*1 #"#"*+& #"*+ 6*'-*%) %# &0 " + '*% + -.#"* )).+#%#*' &#"* *+# -#" ++*# &0 *# $* % +(#" ) ( # +.-%* $#% &*' / % &>& #* & .( $* +.$' / + & +#*+&%& & # %)(*+" ,1"+*/ *#*4+ # &+%*))*#*' & 0 +(*& # &*' *%) *,#"* ) ( #+.-%* %& $* 1 ##*&%+ O B907
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7< .+ &!#"*+%(*$%+ +-.&# &+%+#"%#,#"*'*!**+--**'(*(% &+#"*+%(* %+#"%#0 A&0907%&$*(' >*' &#"*-)) 1 &!(%&&*+#"%# #.+*++%(* $%+ + -.&# &+ %&'+%(* & .( $* -'*!**+ --**'(%+ #"%#0 O M N B90=8C 1"** + #"* ))*# & 1 % / *)*# *Æ *& #+ 1"+* / %).*+ %* 6*, %&' %* #"* +& #"*+ + (%# *+ %&' *+*# / *) 0 "* %$ / * -( .)%# & *)'+ %( .)# *+).# &**+*& #%# &-#"*& #)(*+" $#% &*'/ %&*+.$' / + & +#*&#"**/ .+&#)(*+" 0 "*(*+" +**+*& #*' %+%(*+"%#)*/ *) 8 %)&! 1 #" #"* '*#% ) B1 % / *)*#C *Æ *& #+ %# )*/ *) 80 + (*& # &*' *%) *, #"*+* *Æ *& #+1 .)'$*(*&&26*1"*&#"*+(#") ( #+.-%*'*-(+ / *# (* '.* # +& #"*+ 6*' -* %) %# &0 ( )%)%-#* &* (* +.$' / + & +#*, #"* &*1 & #) (*+" %& $* / *1 *' %+ & #) / *#*4 + # &+ %# )*/ *) 8, 1 % / *)*# *Æ *& #+ %#)*/ *) 8%&' 1 % / *)*# *Æ *& #+ %#)*/ *) , %&' %& $* 1 ##*& %+ O O M N O M N ! B90=C

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7 1"** + #"* ))*# &$%+ + -.&# &+ %# )*/ *)7, , %&' %* #"* & #) / *#*4+ # &+%#)*/ *)7,%&'8*+*# / *) , %&' %*#"*1 % / *)*#*Æ *& #+ %# )*/ *) %&' 8 *+*# / *) , ! + #"* '*& # # (%# 4 %&' + #"* 6* (%# 40 " + '*%%&$*!*&*%) 6*'-%& #)(*+" $#% &*'%-#* 2#"+.$' / + &%&'#"* *+&' &! *4*++ & %& $*1 ##*& %+ O B90=7C 1"** B C OM N, %&' O M N ! ! ! ! ! 0 "* ( .)# *+).# & '&%( + + '*/ *)*' .+ &! #" + -( .)%# & & #"* &*4# +*# &0 90: &%( + "*+(#") ( #+.-%* %&$*(%'*'&%( -#"*& #)/ *#*4+ # &+ %# )*/ *) 8 %&' 1 % / *)*# *Æ *& #+ %# )*/ *) 7 B C %* -.&# &+ # (*0 "* / *) # #"* +(#" +.-%*, & #))*' $ #"* (*+" $#% &*' #".!" +#*+ ('*)+.$' / + &, +! / *& $ S B C O B C S B90==C

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=:Input : Geometry & Topology Information Input : Data for Force Field Synthesis Force Synthesis Module Multilevel Analyzer Multiresolution Analyzer Numerical Analysis Toolkits Update Engine Module Display Finite Element Analysis Module Subdivision Module Topological Information Processing Module !.*<00 +#*( % " #*#.*0

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=9 1 *)) %+ '*#*( &*+ #"* +*# / *# *+ & #"* & # %) (*+" & #)) &! *% " &' / '.%) *)*(*& #0 " +('.)*%)+ / '*+#"* &-(%# &%$.##"*!)$%)(%++,'%( &! %&' +# ?&*++ (%# *+ 1 #".# %#.%)) %++*( $) &! #"*( -( &' / '.%) *)*(*& #%) (%# *+0 <0: *& #"*+ + '.)* "*.+*%&%)/ % .+# *+--*+&#"*+(#") ( #+.-%*0 .*& #) + &! -*+, $%))& -*+, %&' (%!* !%' *& #2$%+*' -*+ %* $* &! +.#*'0 #"* # *+ -*+ %& %)+$* *%+ ) &%#*'0 "* .+* %) *+ + &!-*+ & #"* ) ( # +.-%* $ / ' &! & #+ & = -( 1" " #"* + &!+ &**' # $* %##% "*'0 "*= & #+* >%# & +'&** #"* & #*%# / *)-((.+* &.# -(>)*+0 %)$%))&-*+, #"* .+* &**'+ # +* 1"*#"* #"* $%))& &! -* + # &D%#* #"* ('*) # '*D%#* #"* ('*)0 ).(* (%!*+ %* / '*' %+ &.# # %) (%!* !%' *& #2$%+*' -*+0 '*#% )*' ' +.++ & & #"*+* -* %) %# &(*#"'+ 1 %+ / '*' &*# & =0=0:0 <0; '%#* &! &* -#* #"* .+* / '*+ %)) #"* &**++% &-(%# &%$.# #"* ('*) %&' #"* -* %) %# & (*#"', #"* .'%#* *&! &* &+#%& #) .'%#*+ #"* '&%( +.$2 ' / + & +.-%* ('*) # % " */ * %& *A. ) $ .( + # &0 "* ' +*# 6*' +*&' '* ' ?**& # %) *A.%# & (# & )*%'+ # +)/ &! % )%!* +%+* ) &*% ++#*( *A.%# &+0 &5.!%#* !%' *& # M=:, <;N, %& #*%# / * ) &*% ++#*( +)/ *, + .+*' #" + .+*0 "* .+* %& "+* * #"* ( .)# )*/ *) '&%( + ( .)# *+).# &

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=< '&%( +%% ",%&'#"*.'%#**&! &*%' &!) "+*+ %)! #"(+1"*& #"* '*!**+ -**'(-#"* */ )/ &!+.-%* ('*) + "%&!*'0 <09 +)% '.)* "* .'%#* *&! &* &+#%& #) .'%#*+ #"* '&%( ('*), %&' #"* *)*/ %& # !*(*# %&' #)! %) &-(%# & +%++*' # #"* ' +)% ('.)**&'* &! #"* */ )/ &! +(#" +.-%*0 "* ' +)% ('.)* .+*+ *& !%" + ) $% *+ *&'* &! .+*+0 # %)+ +*/ *+ %+ #"* &.# ('.)* (.+*2$%+*' .+* & #*%# &+0

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"* +*' '&%( -%(*1 3 *4#*&'+ #"* %) %# &%*%+ +.$' / + & +.-%*+$* &'#"*#%' # &%)('*) &!'(% &0 "*-%(*1 3*&"%&*+#"*+. #2 %$ ) # -+.$' / + &+.-%*+-!*(*# ('*) &!0 ##"*+%(*# (*,#"*+*' '&%( ('*)+%&$*.+*'-* / *-+"%*+-(%&!*%&'/ ).(*'%#%,&&2 ! ' (# & *+# (%# &, ( .)# *+).# & *' # &! %&' &%#.%) #*% & ('*) &!0 "* %) %$ ) # #"* '&%( ('*) & #"*+* %*%+ +-.#"* )).+#%#*' & #"* *+# #" + "%#*0 0 *(*# '*) &! "* +*' '&%( -%(*1 3 +.$' / + & +.-%*+ %& $* .+*' # *2 *+*& # % 1 '* / % *# %$ #% #)! %) +"%*+ 1 #" *(%3 %$)* *%+*0 & % # %) !*(*# ('*) &!%) %# &.+ &! '&%( +.$' / + &+.-%*+, #"* .+* %&+* -%& (*+"%+#"* & # %)B& #)C(*+",%&'#"**+&' &!) ( #+.-%* %& $* +.)#*' & #*%# / *) $ %) &! +& #"*+ 6*' -*+0 & '&%( +.$' / 2 + & +.-%* ('*) %& $* .+*' #"* ('*) &! .+*0 "* *4%()*+ +" 1& & #" + +*# & .+* '&%( %#( .))2)%3%&' '&%( (' >*'$.##*D +.$' / + & +.-%* ('*)+0 =

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= !.* 00 B%C, B$C, BC %&' B'C E & # %) +"%*+ B$#% &*' %) &! %#( .))2)%3 +.$' / + &.)*+&& #)(*+"*+CFB*C,B-C,B!C%&'B"CE #"**+&' &!(' >*' +"%*+ %-#* & #*%# / *-* %) %# &0

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:8 +#,'&%( %#( .))2)%3+.-%*('*)+%+*+*& #*' &"%#*=%*.+*' -+"%* ('*) &!0 "* +* %)*)*(*& #+ &#"* ) ( #+.-%* %*'%3)+"%'*' %&' #"* &(%) *)*(*& #+ %* ) !" #) +"%'*' & %)) #"* ('*) &! *4%()*+0 # (% $* &#*' #"%# #"* -( .)%# & *+*& #*' & "%#* ;, 1" " 1 )) "% / * &) &* # * >& #* *)*(*& # & #"* ) ( # +.-%*, %& $* .+*' %+ 1 *))0 & !00, +*/ *%) & # %) +.-%*+$#% &*'-(' ?**& #& #)(*+"*+.+ &!%#( .))2)%3+.$' / + &.)*+ %&'#"**+&' &!(' >*'+.-%*+%-#* & #*%# / *+ &!-*%) %# &%* +" 1&0 "%&!* #"*+"%*-%& & # %)+.-%*, + &!+%*%##% "*'-(' ?**& # & #+ & = # #"* &*%*+# & #+ & #"* ) ( # +.-%* +. " #"%##"* ) ( # +.-%* '*-(+ # 1 %'+ #"*+* & #+ !*&*%# &! #"* '*+ *' +"%*0 & !00B%C, %& *& +.-%* '*>&*' $ %& & # %) (*+" 9 / *# *+ %&' :; -%*+ + +" 1&0 "* (*+" "%+ &* *4#%' &% / *#*4 '*!** ;0 " + ) ( #+.-%* + (' >*'$ %) &! + &! -*+ & #*%# / *) , %&' #"* (' >*' +.-%* + '* #*' & !00B*C0 #.+,'*>&*'$ %& & # %)(*+"-=7/ *# *+%&'=7-%*+,%&' #+(' >*'+"%* + +" 1& & !00B$C%&'B-C*+*# / *) 0 "* & # %)(*+"-#"*+(#") ( #+.-%* +" 1& & !00BC "%+ ;:: -%*+ %&' ;:7 / *# *+, #"*( %* *4#%' &% / *# *+ '*!** ;0 "* ) ( # +.-%* + (' >*' &#*%# / *) $ %) &! + &! -*+ -( / % .+ & #+ & = %&' #"* (' >*' +"%* + '* #*' & !00B!C0 #*#"%##"**4#*& # -'*-(%# &"%+$**& & #*%# / *)& #))*'$ / % &!#"* +# ?&*++ #"* %##% "*' + &!+0 "* .* # & #"* ) ( # +.-%* "%+ $**& '*-(*' $ %) &! + &! -*+ " !"* (%!& #.'*, 1"**%+ #"* ) 1 * # & "%+ $**& (' >*' $ %) &! + &! -*+ ) 1 * (%!& #.'*0 "* +*%' #"*

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: !.*070 B%C,B$C,BC%&'B'CE & # %)+"%*+B$#% &*'%) &!(' >*'$.##*D +.$' / + &.)*+&& #)(*+"*+CFB*C,B-C,B!C%&'B"CE #"**+&' &!(' >*' +"%*+ %-#* & #*%# / *+.)# &!/ %-* %) %# &0 '*-(%# & *?*# + )*%) )%!* & #"* )%##* %+* $ / .+ *%+&+0 &%)) , % D%#+"**# '*>&*' $ %& & # %)(*+"-9:-%*+ %&' / *# *+, +" 1& & !00B'C, + '*-(*' & #*%# / *)#$#% &#"* "%#2) 3 *+"%* +" 1& & !00B"C0 *4#, +"%* ('*) &! *4%()*+ .+ &! #"* '&%( (' >*' $.##*D +.$' / 2 + & +.-%* ('*) %* +" 1&0 "* ) ( # +.-%* "** &+ +#+ % + &!)* # * +(#"# %&!.)%>& #**)*(*& #%# "*+, *+*# / *-#"*& .( $*-*4#%' &% / *# *+ &#"*& #)(*+"0 & !007,+*/ *%) & # %)+.-%*+$#% &*'-(' ?**& #

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:: !.* 0:0 B%C %&!* '%#% %& %& / ) %)&! 1 #" #"* & # %) 6*' ('*), B$C %& & #*(*' %#*+#%!*-*/ ).# &,%&'BC#"*>##*''&%( %#( .))2)%3+.$' / + & +.-%* ('*)0 !.*0;0 B%C %&!*'%#% -% "*%' %)&!1 #" #"* & # %) 6*'('*),B'C #"* >##*' '&%( $.##*D+.$' / + &('*),%&'BC/ +.%) 6%# &-#"*+"%*-(%&#"* / *1 & #0

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:9 !.* 090 B%C +) * -( % $% & , B$C & # %) 6*' ('*) &+ '* #"* *! & & #**+# +.* (+*' & #"* +) *, BC #"* >##*' ('*) +.* (+*' & #"* +) *, %&'B'C%=/ *1-#"*'&%( %#( .))2)%3+.$' / + &+.-%*('*)>##*'# #"* **$*)).(0 !.*0<0 B%C%#% & #+ '*& # &!#"*$.&'%-#"*%.'%#*& .)*.+&% +) *-" .(%&$% &,B$C'%#% & #+B-(%))+) *+C &=%)&!1 #"#"* & # %) 6*' ('*),%&' BC #"* >##*' '&%( $.##*D +.$' / + &('*)0

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: & #"* (%!*0 & # & ..+ (%!*2$%+*' -*+ %+ 1 *)) %+ + &! -*+ %* %) *' # #"*('*)%&'#"*('*)'*-(+.&'*#"* &D.*&*-#"*+*-*+.& # )(%4 ( .( &-(%# &##"*$.&'% *+-#"*'*+ *'%.'%#*+"%*0 "* >&%)>##*' ('*) & = + +" 1& & !00##*' ('*)"%+ =:# %&!.)%-%*+ %&' :/ *# *+0 0= &2 ! '# &+# (%# & "*'&%( +.$' / + &+.-%*('*)+%&$*.+*'*?*# / *)#*+# (%#*#"* (# &-%&&2 ! '$5*#-(%! / *&# (*+*A.*&*-%&!*/ ).(*'%#%0 & *4%()* + +" 1& & !00 1"** #"* (# & #"* )*-#2/ *& # .)% "%( $* % %& &*"*%# +#% 3 *' / *%()*#*%' %)*.+ &!'&%( (' >*'$.##*D +.$' / + & +.-%* ('*)0 "* '%#% +*# ( +*' 9 = (%!*+, 1 #" *% " / ).(* (%!*"% / &!+) *+-7 7 4*)+0 +#,#"*+"%* +* / **'-( &* '%#% +*# .+ &! (%!*2$%+*' B!%' *& #C %+ 1 *)) %+ & #2$%+*' -*+0 + $*-*, #"* & # %) 6*' ('*) "%+ % & #) (*+" ( + &! 7: # %&!.)% -%*+ %&' : / *# *+ 1"**%+ #"* & #) (*+" #"* >##*' ('*) "%+ =: # %&!.)% -%*+ %&' :/ *# *+0 &* #"* +"%* + * / **' -( &* '%#% +*#, #" + >##*' ('*) + .+*' %+ %& & # %) 6%# & #"* &*4# '%#% +*# # #% 3 #"* +"%* & #**+#0 "* +&%+"#+ -((# &#% 3 &!%*+" 1& & !00-#"* 9/ ).(*'%#%+*#+0 # (% $*&#*'#"%##"*& #)(*+"'*+ $ &!#"*+(#"+.-%*++" 1& & !00 "%+ &) =: # %&!.)%-%*+ 1 #" % ##%) : / *# *+ %+ (*& # &*' *%) *0 " + *4* (*& # )*%) +" 1+ #"%# . ('*) %& $* .+*' # #% 3 % +"%* & #**+#

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: !.*00 &%+"#+-(#"*%& (%# &-%& &*"*%#(# & / *%%' %)* .+ &! #"* '&%( $.##*D +.$' / + &('*)0

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;8 !.* 00 B%C "* & # %) 6*' ('*) %)&! 1 #" #"* '%#% +*#F B$C #"* >##*' ('*) 1 #" # 1 +.$' / + &+ & #"* & # %) (*+", %)&! 1 #" %##% "*' + &!+ *' # &!0 "* ('*) %-#* *' # &! BC %# ) 1 * *+).# &, B'C %# #"* +%(* *+).# & #"* >##*' ('*),%&'B*C %#" !"**+).# &0 -( % +*# # (*'**&'*& # / ).(* '%#% +*#+ & %& *Æ *& # (%&&*0 #* #"%# & #"**4 +# &!.*)!*(*# +.$' / + &+.-%*#* "& A.*%&$*.+*'1 #"B# (* / % &!C & # & ..+ '%#%+*#+0 0: .)# *+).# &' # &! "*+*''&%( +.$' / + &#* "& A.*+ *+*& #" *% " %)+"%** /2 * %&' +"%* ('*) &! 1 #" & % ((& -%(*1 3, 1"** #"* ('*)* %& +%& &='%#% & #+-%## *('*),* / *#"*.&'*) &!+"%*-(#"* & #

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