Group Title: Department of Computer and Information Science and Engineering Technical Reports
Title: Dynamic Catmull-Clark subdivision surfaces
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Title: Dynamic Catmull-Clark subdivision surfaces
Series Title: Department of Computer and Information Science and Engineering Technical Report ; 97-020
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Language: English
Creator: Qin, Hong
Mandal, Chhandomay
Vemuri, Baba C.
Affiliation: University of Florida
University of Florida
University of Florida
Publisher: Department of Computer and Information Science and Engineering, University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: November 10, 1997
Copyright Date: 1997
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Dynamic Catmull-Clark Subdivision
Surfaces



Hong Qin Chhandomay Mandal Baba C. Vemuri





CISE Technical Report # TR-97-020

Department of Computer and Information Science and Engineering

P.O. Box 116120

University of Florida, Gainesville, Florida 32611.

Email: {qin Icmandal I vemuri}'-i.-, .ufl.edu
Tel: (352) 392-1482, (352) 392-1220 (fax).
10th November, 1997


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Abstract


Recursive subdivision schemes have been extensively used in computer graphics, computer-

aided geometric design and scientific visualization for modeling smooth surfaces of arbitrary

'I" 'I -. Recursive subdivision generates a visually pleasing smooth surface in the limit from an

initial user-specified i" ..1- i ,1 mesh through the repeated application of a fixed set of subdivision

rules. In this paper, we present a new dynamic surface model based on the Catmull-(' I !:

subdivision scheme, which is a very popular method to model complicated objects of arbitrary

genus because of 11ii i- of its nice properties. Our new dynamic surface model inherits the

attractive properties of the Catmull-( l: subdivision scheme as well as that of the I'! -', --

based models. This new model provides a direct and intuitive means of manipulating geometric

shapes, a fast, robust, and hierarchical approach for recovering complex geometric shapes from

range and volume data sets using very few degrees of freedom (control vertices). We provide

an :lii 11- I, formulation and introduce the i'1! -i. 1 quantities required to develop the dynamic

subdivision surface model which can be interactively deformed by ;,1, 1- ii,- synthesized forces in

real time. The governing dynamic differential equation is derived using Lagrangian mechanics

and a finite element discretization. Our experiments demonstrate that this new dynamic model

has a promising future in computer graphics, geometric shape design and scientific visualization.


Keywords

Computer Graphics, CAGD, Visualization, Subdivision Surfaces, Deformable Models, Dy-

namics, I !ii!I- 1.I ii! I- Interactive Techniques.


I. INTRODUCTION

Generating smooth surfaces of arbitrary I. 1, .1- .- is a grand challenge in geometric modeling, computer

graphics and visualization. The recursive subdivision scheme first introduced by ('I ,ii:ii [1] is very well

suited for this purpose. During the past two decades, a wide i of subdivision schemes for modeling

smooth surfaces of arbitrary , .1. - have been derived in geometric modeling after ( '!I i pioneering

work on the curve generation. A recursive subdivision algorithm I i' 11 generates a smooth surface


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which is the limit of a sequence of recursively refined polyhedral surfaces based on a user-defined initial

control mesh. At each step of the subdivision, a finer polyhedral surface with more vertices and faces will

be constructed from the previous one via a refinement process (also called i'ipp'!i < .! 'i ). In general,

subdivision schemes can be categorized into two distinct classes namely, (1) approximating subdivision

methods and (2) interpolating subdivision methods.


A. Background

Among the approximating schemes, the techniques of Doo and Sabin [2], [3], [4] and Catmull and ('I! :

[5] generalize the idea of obtaining biquadratic and bicubic B-spline patches from rectangular control

meshes. In [5], Catmull and ('I !: developed a method for recursively generating a smooth surface from

a polyhedral mesh of arbitrary 1 1- .1,_ -. The Catmull-('I !: subdivision surface, defined by an arbitrary

non-rectangular mesh, can be reduced to a set of standard B-spline patches except at a finite number

of extraordinary points, where the in-degree of the vertex in the mesh is not equal to four. Doo and

Sabin [3] further :i!I .1- .1 the smoothness behavior of the limit surface near extraordinary points using

Fourier transforms and an eigenvalue :;i i1- -i- of the subdivision matrix. Ball and 1I ,*i [6], [7] and Reif

[8] further extended the prior work on !il li- properties of subdivision surfaces by deriving various

necessary and itL!!i, !0 conditions on smoothness for different subdivision schemes. In [9], Loop presented

a similar subdivision scheme based on the generalization of quartic triangular B-splines for triangular

meshes. Halstead, Kass and Derose [10] proposed an algorithm to construct a Catmull-('I! !: subdivision

surface that interpolates the vertices of a mesh of arbitrary 1 I 1- _1 -. In [11], Taubin developed a signal

processing-based approach to fair polyhedral surfaces of arbitrary l .1 1.._ .

The most well-known interpolation-based subdivision scheme is the "1. 111. il algorithm proposed by

Dyn, Gregory and Levin [12]. Butterfly subdivision method 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 for the initial polygonal meshes in order to obtain a smooth limit

surface. A variant of this scheme was proposed by Dyn, Hed and Levin [13]. Recently, Zorin, Schroder

and Sweldens [14] further developed an improved interpolatory subdivision scheme that can retain the


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-ii! 1.ii '1i of the butterfly scheme and result in much smoother surfaces even from initial polygonal meshes

that are irregular.


B. Motivation

Although recursive subdivision surfaces are extremely powerful to represent smooth geometric shapes of

arbitrary I .1- .,_ -, they constitute a purely geometric representation, and furthermore, conventional geo-

metric modeling with subdivision surfaces 11i be infeasible 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 I | 1, 1

spline-based modeling schemes. In addition, it 11 ,- not be enough to obtain the most ''I !! surface that

interpolates a set of (ordered or unorganized) data points. A certain number of local features such as

bulges or inflections ("i. o' .i!. -- ') ii ,- be strongly desired while making geometric objects satisfy global

smoothness requirements in geometric modeling and graphics applications. In contrast, 1,1i -i. --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

1I1!- -., 1 laws are of particular interest in this context. These models respond dynamically to applied

forces in a very intuitive manner. The equilibrium state of the model is characterized by a minimum of

the potential i. of the model -,L,.j. I to imposed constraints. The potential ii. i functionals can

be formulated to satisfy local and global modeling criteria and impose geometric constraints relevant to

shape design.

Free-form deformable models were first introduced to computer graphics and visualization in Terzopou-

los et al. [15] and further developed by Terzopoulos and Fleischer [16], Pentland and \\ ill !!- [17],

Metaxas and Terzopoulos [18] and Vemuri and I ,l1- li. 1i [19]. Celniker and Gossard [20] developed

a -* -, i i for interactive free-form design based on the finite element optimization of. i I function-

als proposed in [16]. Bloor and \\ !I-. ., [21], [22], Celniker and Welch [23] and Welch and \\ i1!: i [24]

proposed deformable B-spline curves and surfaces which can be designed by imposing the shape crite-

ria via the minimization of the i. functionals -11lj. i I to hard or soft geometric constraints through


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Lagrange multipliers or i,. ii 11 methods. Recently, Qin and Terzopoulos -'2], [26], [27] have developed

dynamic NURBS (D-NURBS) which are very sophisticated models suitable for representing a wide i 1I

of free-form as well as standard :I ,! I1- I shapes. The D-NURBS have the advantage of interactive and

direct manipulation of NURBS curves and surfaces, resulting in 1,1i -11 1-! meaningful hence intuitively

predictable motion and shape variation.

A severe limitation of the existing deformable models, including D-NURBS, is that they are defined

on a parametric domain. Hence, it is almost impossible to model surfaces of arbitrary genus using these

models. In this paper, we develop a dynamic generalization of recursive subdivision schemes based on

Catmull-( 'I i I: subdivision surfaces. Our new dynamic model combines the benefits of subdivision surfaces

for modeling arbitrary l- ., .1- _- as well as the dynamic splines for direct and interactive manipulation of

shapes by ,ii, 1-i i1- simulated forces. Note that, the derivation of our dynamic subdivision surface poses a

significant technical challenge because of the fact that no closed-form parameterization of the limit surface

exists near the extraordinary points. We present the details of our formulation in a later section.

The dynamic Catmull-('I i!: subdivision surface has been developed primarily for modeling arbitrary

,l '.1,1--. However, another important application of the developed model is in shape recovery. In a

S- I,'. ,1 shape reconstruction application, we need to recover shapes of arbitrary l. I 1-_-- from large data

sets. P! -i, --based models are often used for this purpose. However, the model used for fitting should

be able to recover the shape accurately. At the same time the number of degrees of freedom for model

representation should be kept low. Another important criterion is that the model initialization should

not be restricted to parameterized input meshes since it is infeasible to parameterize shapes of arbitrary

, ,! 1,,_-. A 1.1! -1 --based model ;ri-f' ;-_ the aforementioned criteria is a good candidate for a solution

to the shape recovery problem.

Pr! -i, --based deformable models used to solve shape recovery problem involve either fixed size [19], [28],

[29], [30], [31] or adaptive size [32], [33], [34], ,;], [36], [37] grids. The models with fixed grid size generally

use less number of degrees of freedom for representation, but the accuracy of the recovered shape is lacking

in ii i- cases. On the other hand, the number of degrees of freedom used for shape representation of

the model is generally very high and computationally expensive ad hoc schemes are used in models with


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adaptive grid size methods. The recovered shape is however satisfactory in the context of accuracy. The

hierarchical shape representation using locally adaptive finite elements discussed in [34] can ti1. i0 I;

represent the shape of an object of genus zero with a small number of nodal points. However, this scheme

can not be easily extended to cope with arbitrary shapes. The balloon model for describing the shape

of complex objects [32] also adapts the mesh surface to local surface shapes and is purely driven by an

applied inflation force towards the object surface from the interior of the object. This scheme involves

a large number of nodal points for representing complex shapes. Moreover, all the existing models using

either a fixed or an adaptive grid size require a parameterized mesh as their input.

The proposed model solves the shape recovery problem very tin. wi 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 ;~,i- arbitrary input mesh, not necessarily parameterized, with an arbitrary number

of extraordinary points. The initialized model deforms under the influence of synthesized forces to fit

the data set by minimizing its ii. i 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 prescribed

error criteria for fitting the data points is achieved.

In a nutshell, the dynamic Catmull-('I !!: subdivision surface model has been motivated by its I '111-

to model arbitrary 1..1-1.._-** where modelers can directly manipulate the smooth limit surface in an

intuitive fashion and by its applicability to the shape recovery problem.


C. Overview

The rest of the paper is organized as follows: Section II presents the detailed formulation of the dynamic

Catmull-( 'I i I: subdivision Surfaces. The implementation details are provided in Section III. Experimental

results can be found in IV. Ii II ,11 !, we make concluding remarks and point out future directions of research

in Section V.


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II. FOI:. II NATION


In this section we present a -I in, i i formulation of our new dynamic model based on Catmull-( 'I i!

subdivisions. 1 i -i1 we ',i fi review the Catmull-('I! I: subdivision scheme. Then, we demonstrate how

to assign a bicubic patch in the limit surface to a non-boundary face in a rectangular setting. We further

generalize this idea to assign the infinite number of bicubic patches in the limit surface to faces that are in

the i! ii of an extraordinary point/vertex. Next, we formulate a closed form ;,!i ,- 1 i, 1 representation

of the limit smooth surface which can be viewed as a function of its (initial) polyhedral control vertices.

I i i!! ,11 we introduce ,1i -1. i1 quantities into our dynamic model in order to derive its motion equation.


A. Catmull-(l I subdivision .... f....

Catmull-('I !: subdivision scheme, like ;,i, other subdivision scheme, starts with a user-defined mesh

of arbitrary l, ,1, .1, _- 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 sequence of recursively refined

polyhedral meshes will converge to a smooth surface. The subdivision rules are as follows:

For each face, introduce a new face point which is the average of all the old vertices defining the face.

For each (non-boundary) edge, introduce a new edge point which is the average of the following four

points: two old vertices defining the edge and two new face points of the faces .'li 1 ,!I to the edge.

For each (non-boundary) vertex, introduce a new face point obtained from the average -F+ 2-iE+ (i-3

where F is the average of the new face points of all faces ,li I !i to the old vertex point, E is the

average of the midpoints of all edges incident on the old vertex and n is the number of the edges

incident on the vertex.

Form new edges by connecting each new face point to the new edge points of the edges defining the

old face and by connecting each new vertex point to the new edge points of all old edges incident on

the old vertex point.

Define new faces as those enclosed by new edges.

The most important property of Catmull-(' Il: subdivision surfaces is that the smooth surface can

be generated from control meshes of arbitrary I ,1i .1. .- *. Therefore, this subdivision scheme is extremely


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valuable for modeling various complicated geometric objects of arbitrary l. 1. 1,_- -. Catmull-('I I: subdi-

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

tensor-product B-spline surface for a rectangular control point mesh). In addition, the aforementioned sub-

division rules generalize the recursive 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 extraordinary

points. Note that, after the first subdivision, all faces are quadrilaterals, hence all new vertices created

subsequently will have four incident edges. The number of extraordinary points on the surfaces remains a

constant which is determined by the refined meshes after one subdivision. The limit surface is curvature-

continuous everywhere except at extraordinary vertices, where only tangent plane .i0ii,,i I is achieved.

In spite of the popularity of Catmull-( 'I i!: subdivision surfaces for representing complex geometric shapes

of arbitrary I. 1 i .1. -, these subdivision surfaces are not parameterizable and lack closed-form i,! i1- i for-

mulations. These deficiencies preclude their immediate pointwise manipulation and hence 11i i restrain

the applicability of these schemes. We develop a new dynamic model based on Catmull-( 'I !: subdivision

surfaces which offer modelers a closed-form I, 1 i,1 111 formulation and allows users to manipulate the model

directly and intuitively.

To develop the dynamic model which treats the limit smooth surface as a function of its control mesh in

a hierarchical fashion, we need to update control vertex positions continually at ;,i!- given level. However,

all the vertices introduced through subdivision are obtained as an ;.!!i.- combination of control vertex

positions of the initial mesh. Therefore, we can control the dynamic behavior of the limit surface by

formulating 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. 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

subsections, we present a scheme of assigning the bicubic patches to various faces of the initial mesh. It

11i be noted that one additional subdivision step 11i be needed in some cases to isolate the extraordinary

points and treat the obtained mesh as the initial mesh (one I- 1. ,1 example is when the initial mesh is a

tetrahedron).


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B. Assigning patches to regular faces


------------------------------------------------
(-----------i

Fig. 1. A rectangular mesh and its limit surface consisting of 4 bicubic surface patches.



In Fig.1, 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 valence of 4 at each vertex, the

corresponding bicubic surface patch can be assigned to it in a straight forward way. In Fig.1, the surface

patches S1, S2, S3 and S4 are assigned to face F1, F2, F3 and F4 respectively. The 16 control points for

the patch SI, corresponding to face F1, are highlighted in Fig.l.


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S
2 \
A













SS 6
---- ------ I---------- -- -
S' i S4 i




__-- -- _---- -- -











Fig. 2. A mesh with an extraordinary point of valence 3 and its limit surface.


C. Assigning patches to irregular faces


In Fig.2, a mesh containing an extraordinary point of valence 3 and its limit surface are shown. The

faces Fo, FI, ..., F8 are assigned to bicubic patches So, S1,..., S8 respectively (as they all have vertices

of valence 4) following the aforementioned scheme. However, the central smooth surface enclosed by the

patches So, S,..., S8 consists of infinite number of bicubic patches converging to a point in the limit. We

need to develop a recursive way of enumerating these bicubic patches and assigning them to various faces

at -!t.11 i. Ii levels in order to develop the dynamic subdivision surface model.

The idea of enumerating the bicubic patches corresponding to faces having an extraordinary vertex

is shown in 1 i ; where a local subdivision of the mesh consisting of faces F0, F, ..., F8, P, P1, P2 (and

not the other boundary faces) of Fig.2 is carried out. Topologically, the resulting local subdivision mesh

(shown as dotted mesh) is exactly the same as the mesh in Fig.2 and hence exactly the same number of


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selected mesh for
local subdivision










P S













nn
n n
HR













A\
\\
n n

n n
\ \
i,-''
-- -- -- -- -- -- -- -- -




nf











Fig. 3. Local subdivision around the extraordinary point and the limit surface.


bicubic patches can be assigned to its faces with vertices of valence 4 as is evident from 1 !, ; (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 steps. This description is

for formulation purposes only and the exact implementation will be detailed in a later section.



D. Kinematics of the limit .... f..,


In this section we develop the mathematics for the kinematics of the limit surface via illustrative


examples and then present the generalized formulas. We start the illustration with a single bicubic B-


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spline patch which is obtained as the limiting process of the Catmull-( 'I 1i subdivision algorithm applied

to an initial 4 by 4 rectangular control mesh. Let sp(u, v), where (u,v) E [0,1]2, denote this bicubic

B-spline patch which can be expressed;! 1- 11. as
3 3
s p(u, v) = (x(u, v), y(u, v), z(u, v)) = di,jBi,4(u)Bj,4(v) (1)
i=0 j=0
where dij 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. The subscript p on s denotes the patch under consideration.

Expressing Eqn.1 in a generalized coordinate -- -. ii we have

Sp = Jpqp (2)

where Jp is the standard Jacobian matrix 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 = 4i +j respectively in the vector

qp. We can express the entries of Jp explicitly in the following way: Jp(0, k) = Jp(1, k+1) = Jp(2, k+2) =

Bi,4(u)Bj,4(v) and Jp(0, k + 1) = Jp(0, k + 2) = Jp(1, k) = Jp(1, k + 2) = Jp(2, k) = Jp(2, k + 1) = 0.


D.1 Limit surface with i ii i bicubic patches from a rectangular initial mesh

Now let's consider a limit surface consisting of i ,ii~- bicubic surface patches obtained after .'1,1- i,!

an infinite number of subdivision steps to a rectangular initial mesh. For example, let the limit surface of

Fig.1 be s, which can be written as
1 1 1 1
s,, (u, v) =s, (2,, 2v) + s, (2(u ), 2v) +s,, (2(u ), 2(v- -)) +s ( 2(v )) (3)

where s,, (.,, 2v) = s,,(u, v) for 0 < u, v < and 0 otherwise. Similarly, s-,, S,I3 and s,4 are also equal

to s, (u, v) for an appropriate range of values of u, v and 0 outside. It i1 be noted that sM, s,,, Ms,, s, 4

correspond to patches S1, S2, S3, S4 respectively in Fig.1. Rewriting Eqn.3 in generalized coordinates we

have
4
s, = Jiqi + J2q2 +J3q3 + J4q4 = Jii (4)
i=1


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where Jis are the Jacobian matrices of size (3,48) and qis are the (x,y,z) component concatenation of a

subset of the control points of s, defining s,,, i = 1, 2, 3 and 4. A more general expression for s,, is
4
sm = JiAlqm + J2A2qm + J3A3qm + J4A4qm = JiAiqm = Jrqm. (5)
i=1
Where, q, is the 75-component vector of 3D positions of the 25 vertex control mesh defining the limit

surface s,.n Matrices 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, = i=l JiA.


D.2 Limit surface with i! i ,- bicubic patches from an arbitrary initial mesh

The stage is now set to define the limit surface s using the vertices of initial mesh 4M for i~- arbitrary

1..**I' ,!_-, assuming all faces are rectangular 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 11i be required and the resulting mesh can be treated

as the initial mesh. Let the number of vertices in the initial mesh 4M be a, and let I of these be the

extraordinary vertices. Let us assume that the number of faces in the initial mesh are b, and that k of

these have vertices with valence 4 (henceforth termed a "-ii.. i I I f ) and each of the remaining (b k)

faces have one of the I extraordinary vertices (henceforth termed a "-1 i '! i ). Let p be the 3a = N

dimensional vector containing the control vertex positions in 3D. Using the formulations in subsections

II-B and II-C, the smooth limit surface can be expressed as
k I
s= n + S (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 num-

ber of bicubic patches corresponding to each of the extraordinary points. "',i '1~,!'! ; the same approach

taken before to derive Eqn.5, it can be shown that
k k k
>ni = E("Ji)("pi)= ( '("Ji)("Ai))p = ("J)p (7)
i=1 i=1 i=1
where "Ji,"pi and "Ai are the equivalent of Ji,pi in Eqn.4 and Ai in Eqn.5 respectively. The pre-

superscript n is used to indicate that these mathematical quantities describe bicubic patch in the limit

surface corresponding to normal faces.

We will use the following notational convention for describing various mathematical quantities used


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in the derivation of the expression for a collection of infinite number of bicubic patches around an ex-

traordinary vertex. The 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 !I i0 I- and the level indicator (to represent various levels of

subdivision in the local control mesh around an extraordinary vertex) is depicted via subscripts on the

curly braces.

The expression for sj is derived using the recursive nature of local subdivision around an extraordinary

vertex as shown in subsection II-C. i -1, sj can be expressed as

sj = {'Jj}1{SP}I + {sj} (8)

where the first term of Eqn.8 is the generalized coordinate representation of the bicubic B-spline patches

corresponding to the normal faces of the new local subdivision mesh obtained after one subdivision step

on the local control mesh (similar to those patches marked n in Fig.3). {sjy} 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). The vertices in the newly obtained local 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 subdivision) following the subdivision rules. We can name this subset of initial control vertices

{ PJ}0. Furthermore, there exists a matrix {"Bj}1 of size (3c,3d), such that {8Bj5}{5pj,} = {5pj}i

where {Spj}l and {fpj}o are vectors of dimension 3c and 3d respectively. AI.I1. i,_ the idea of recursive

local subdivision again on {sy},, sy can be further expanded as

Si = {SJj}l{SBj}j{spj}o + {'SJj}2{SB}2 }1 + {sj}2 (9)

In the above derivation, {'y I}, is a vector of dimension 3d, comprising of a subset of the vertices defining

the 3c dimensional vector {5py}1. Note that, {fpj}j has the same structure as {pjjY}, therefore, there

exists a (3d, 3d) matrix {"Cjy} such that {'Cjy 1} pjy = {SP} 1l. 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. So, if we proceed one step further, we obtain

s e = {SJth iSBt}i spj}o + fJj2 t 2e cl r 'J B 2 fCu p s 0 + 3' arJ B Cd t3 e oI {s} (10)j

Because of the intrinsic property of the local recursive subdivision around the extraordinary point, we


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have {SJj = {,= {J = }2 = {Jj} = ... = {'Jj} In addition, the subdivision rules remain the same
throughout the refinement process, we also have {"Bj}1 = {"Bj}2 ... = {"B},2 = ...= { B3j}. So,
we can further simplify the above equations leading to

sj = {SJ ll SIBll JSpj}o + SJll S }Jll SCJl Sp o + SJll S } JSC l p }o ...

O-o
i= 0


0 -,,


{scj}1 {s 0
i I fpo


][ 12


s
fScjI f}2 s?}


{ ][3}3


Fig. 4. Local subdivision around the extraordinary point and the corresponding patches in the limit
surface from -I,!!. i. 1 levels of subdivision.


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We can rewrite sj as

si = (SJj)(SPj) (12)

where SJj = {SJj }{SB }I,(' {sCj/}) and "pj = {SP/ }0* The idea of local recursive subdivision

around an extraordinary point is illustrated in Fig.4. Note that, each vertex position in the subdivided

mesh is obtained by an ;~!hil- combination of some vertices in the previous level and hence ;~i- row of

{"Cj } 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 [10]. The rest of the derivation leading to

an expression for s is relatively straight forward. Using the same approach used to derive the Eqn.7, it

can be shown that

sj = (Jj)(Spj) = ( (SJj)(SAj))p = (SJ)p (13)
j=1 j=1 j=1
From Eqn.6,7 and 13,

s = ("J)p + (SJ)p (14)

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

s = Jp (15)


E. D ,,.. .. ',.

We now treat the control point positions (alternatively, the vertex positions in the initial mesh) defining

the limit surface s as a function of time in order to develop our new dynamic model. The velocity of the

surface model can be expressed as

s(u, v,p) = Jp (16)

where an overstruck dot denotes a time derivative. The 1 11! -i. of the dynamic subdivision surface model

is based on the 1- i -_ version of Lagrangian dynamics [38] and is formulated in an analogous way

to that in [27].

In an abstract 1,1i 1 -- in let pi(t) be a set of generalized coordinates which are functions of time

and are assembled into the vector p. Let fi(t) be the generalized force assembled into the vector fp and

acting on pi. The Lagrangian equation of motion can then be expressed as

Mpi + Dp + Kp = fp (17)


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Let pt(u, v) be the mass 1. i!-i function of the surface. Then

M= ff IjJdudv (18)

is an N x N mass matrix. Similarly the expression for damping matrix is

D= JJJ Jdudv (19)

where -(u, v) is the damping 1L i!.i-I

A thin-plate-under-tension i model [39] is used to compute the elastic potential !, i of the

dynamic subdivision surface. The corresponding expression for the stiffness matrix K is

K = (a nJJu + 22 v + 11J JJu +12 JTu 22J vJ)ddudv (20)

where the subscripts on J denote the parametric partial derivatives. The aii(u,v) and ij (u, v)s are

S1 -I i. i functions controlling local tension and i_-ii I- in the two parametric coordinate directions. The

generalized force vector fp can be obtained through the principle of virtual work [38] done by the applied

force distribution f(u, v, t) and can be expressed as

fp= Tf(u, v, t)dudv (21)


E.1 Multilevel Dynamics

Our dynamic Catmull-( 'I I: 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.15 and Eqn.16) in the dynamic -- -I. I, will be replaced by a larger number of

degrees of freedom q, where q = Ap. A is a global subdivision matrix of size (M, N) whose entries are

uniquely determined by Catmull-('I I: subdivision rules (see Section II-A for the details about the rules).

Thus, p, expressed as a function of q, can be written as

p = (ATA) ATq = Bq (22)

where B = (ATA) 'AT. Therefore, we can rewrite Eqn.15 and Eqn.16 as

s = (JB)q (23)

and

(u, v, q) = (JB) (24)


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respectively. Now we need to derive the equation of motion for this new subdivided model involving a larger

number of control vertices namely q. We need to recompute the mass, damping and stiffness matrices

for this "-- i level. The structure of the motion equation as given by Eqn.17 remains unchanged, but

the l i1 ,. 1!-i..i il 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

Mqi + Dq + Kqq = fq (25)

where Mq = f /pBTJTJBdudv and the derivation of Dq, Kq and f, follow suit.

It ii ,- be noted that further subdivision, if necessary, can be carried out in a similar fashion. Therefore,

multilevel dynamics is achieved through recursive subdivision 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 design requirements. Alternatively, the -- -I. i. can automatically determine the level of

subdivision most suitable for an application depending on some application-specific criteria.


III. FINITE 1.1.1.. I.NT I. IP1.1.. I.NTATION


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.17. An iii1 1- I, G1 solution of the governing

differential equation can not be obtained in general. However, an. mtn. ii numerical implementation can

be obtained using finite element ; !! ,1- -1- techniques [40]. For the dynamic subdivision surface model, two

I of finite elements are considered 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). 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 subsections.


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A. Data S, '. ,.,. .*


A subdivision surface defined by a control mesh at ;,i- 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. The 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 .l !. i. !!I levels. 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.


A.1 Normal i.. !,. !i -

Each normal element is a bicubic surface patch and is hence defined by 16 vertices (from the 8-connected

neighborhood of the corresponding normal face). 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 i!! i,- out

the necessary integration .1 ,!- II i.11 The matrix J in Eqn.18,19 and 20 needs to be replaced by Jp

(of Eqn.2) for computation of the local M, D and K matrices respectively of the corresponding normal

element.


A.2 Special i. !i -


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 II-C. 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 = Nf I !0 (26)


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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 QS 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 an exactly similar fashion.

Now, these mass, damping and stiffness matrices can be assembled to form the mass, damping and

stiffness matrices of the special element. As mentioned in Section II-D.2, the infinite series summation is

convergent. However, it has been found that the contribution from bicubic patches of a special element

at a higher level of subdivision to the mass, damping and stiffness matrices becomes negligible and in the

implementation, the local subdivision is carried out until the contribution is small enough to be ignored.


B. Force Application

The force f(u, v, t) in Eqn.21 represents the net effect of all applied forces. The current implementation

supports spring, inflation as well as image-based forces. However, other I- ,. 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 (uo, Vo) on

the limit surface, the net applied spring force being

f(u, v, t) = k(do s(u, v, t))(u- ,o,v vo)dudv (27)

where 6 is the unit impulse function i1 i11- f(uo, v, t) = k(do s(uo, v0, t)) and vanishes elsewhere in

the surface. However, the 6 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 a mouse button or

the points from which forces need to be applied can be read in from a t!.

To recover shapes from 3D image data, we synthesize image-based forces. A 3D edge detection is

performed on a Gaussian smoothed volume data set using the 3D Monga-D(, i !,. .II)) operator [41] 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
VP(x, y, z)
f(x,y,z)= k P (28)
H VP( x, y, z) z
where k controls the strength of the force. The applied force on each element is computed using Gaussian


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quadrature for evaluating Eqn.21 in Cartesian coordinates. It !i be noted that we can apply spring

forces in addition with the image-based forces by placing points near the region of interest in the slices of

the 3D image data.


C. Discrete D., ...... Equation

The 1!l, i- i!!ii i equation given by Eqn.17 is integrated through time by discretizing the time derivative

of p over time steps At. The 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

p(t + At) 2p(t) + p(t At) (29)
j(t+ At) = A2()
At2

and

(t At) = p(t + At) p(t At) (30)
p^(+ At) = (30)
2At
Using Eqn.17,29 and 30, the discrete equation of motion is obtained as

(2M + DAt + 2At2K)p(t + At) = 2At2fp(t + At) + (DAt 2M)p(t At) + 4Mp(t) (31)

This linear -I. i 1 of equations is solved iteratively between each time step using the ..I il- il.- gradient

method. For a first order -- -1. i, with no mass, the above equation reduces to

(D + 2AtK)p(t + At) = 2Atfp(t + At) + Dp(t At) (32)

which gives a faster convergence.


D. Model Subdivision

The initialized model grows dynamically according to the equation of motion (Eqn.17) and when an

equilibrium is achieved at a given level of subdivision, the model can be subdivided, if necessary, according

to the Catmull-('I I: subdivision rules to increase the number of vertices (control points) and a better fit

to the data can be achieved. Currently the error of fit criteria is based on distance between the data points

and the points on the limit surface where the corresponding springs are attached. However, other I- i" -

of error criterion can also be defined and used in this context. For example, in the context of image-based

forces, if the model. i does not change between successive iterations indicating an equilibrium for the

given resolution, the model can be subdivided further until the model !!. i -_- is -it!l, !. !,I I1 small and the


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change in i i- between successive iterations becomes less than a pre-specified tolerance.


IV. RESULTS

The proposed dynamic subdivision surface can be used to represent a wide ,i. of shapes with

arbitrary genus. In this section we demonstrate the power of our modeling scheme via model fitting

examples to a i i. of data sets of ii,- degree of,! !.ii1 -.i In all the experiments, normal elements

are shaded yellow, while special elements are colored green.

In i -" .) an open limit surface defined by an initial mesh of 61 vertices and 45 faces is shown. The

mesh has one extraordinary point of valence 5. The limit surface is acted upon by spring forces as shown

in I 'i, .(b). The evolving model and its control mesh is shown in Fig.5(c) and(d). The final fitted model

is depicted in Fig.5(e) and (f). It 1i i- be noted that the model controlled by the initial mesh reached local

minimum without fitting the points exactly. In order to obtain an exact fit ( i, '-(f)), the control mesh

is subdivided once thereby increasing the degrees of freedom (control vertices) of the ,11 L!. 1 i! model.

Thus the dynamics can be applied in a hierarchical fashion. The developed model can be used to obtain a

very fast approximate fitting with fewer number of vertices and an exact fit after more subdivision steps

as needed.

In the next experiment, we show the fitting process using spring forces with a closed surface of genus

two(Fig.6). The smooth surface is controlled by an initial mesh of 544 faces and 542 vertices, 8 of them

being extraordinary points of valence 5. In this experiment, the model has -it!, !. !!, degrees of freedom

and fitted the data points exactly without needing further subdivision of its control mesh.

In all the experiments to follow, the initialized model had 96 faces and 98 vertices, 8 of them being

extraordinary vertices of valence 3. The final fitted model, obtained through one step of subdivision, has

a control polygon of 384 faces with 386 vertices. The tolerance level of the error in fit, which is defined

as the maximum distance between a data point and the nearest point on surface as a percentage of the

object diameter, was set to be 1 .

In Fig.7, we demonstrate the model fitting algorithm applied to laser range data acquired from multiple

views of a light bulb. Prior to ,,I'1 i,,- our algorithm, the data were transformed into a single reference


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coordinate -- -. i!, The model was initialized inside the 1000 range data points on the surface of the bulb.

In the next experiment, the shape of a human head is recovered from a range data set as shown in

1i, ', The range data set has 1779 points in 3D. It ii be noted that the final shape with a very low

error tolerance is recovered using very few number of control points in comparison to the number of data

points present in the original range data set. I II!!, example with an anvil data set is shown in Fig.9.

The anvil data set has 2031 data points.

We show the application of our model to anatomical shape recovery from 3D volumetric 1, li1 data in

the last two experiments. 1 i we fit the model to a cerebellum (a cortical structure in brain) given an

input of 30 sagittal slices from a I: 1 brain scan. Fig.10(a) depicts a slice from this 1: i1 scan and the

model initialization is shown in Fig.10(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. Fig.10(c) depicts an intermediate stage of the model evolution during the

fitting process and the final fitted model is shown in Fig.10(d). Arbitrary 3D views of the fitted model

from different viewing angles are depicted in Fig.10(e) and (f).

In the last experiment, we present the shape extraction of a caudate nucleus (another cortical structure

in human brain) from 64 -I l1 I slices, each of size (256, 256). Fig.11(a) depicts a slice from this I- I I scan

along with the points placed by an expert neuroscientist on the boundary of the shape of interest. Fig.11(b)

depicts the data points (placed in each of the slices depicting the boundary of the shape of interest) in 3D.

Note that points had to be placed on the boundary of the caudate nucleus due to lack of image gradients

delineating the caudate from the surrounding tissue in parts of the image. Fig.11(c) depicts the initialized

model and the data points. Continuous image based forces as well as spring 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 caudate shape. Fig.11(d) depicts an intermediate stage of the model evolution

during the fitting process and two arbitrary views of the final fitted model in 3D is shown in Fig.11(e)

and (f).


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V. CONCLUSIONS


In this paper, a dynamic generalization of the Catmull-('I l: subdivision surfaces is presented which

has numerous applications in geometric modeling, computer graphics and visualization. Apart from

providing a direct and intuitive way of manipulating shapes, it facilitates the modeling and shape ;i! I1 -i-

of objects contained in range and volume data sets using very few degrees of freedom. We have presented

an ;li 1- 11, formulation of the subdivision scheme, incorporated the advantages of free-form deformable

models in subdivision scheme, introduced hierarchical dynamic control and shown the advantages of our

model via experiments. However, the current scheme can not recover very sharp edges in the data. Also,

the initialization is interactive; ideally, initialization should be done automatically on the basis of the

input data set. Our future efforts will be focused toward addressing these issues.


VI. ACKNOWLEDG(I.. !1.NTS


This research was supported in part by the NSF grant ECS-9210648 and the NIH grant RO1-LM05944

to BCV, the NSF ('.\il:.i.i award CCR-9702103 and DMI-9700129 to HQ. We also wish to acknowledge

Dr. H. Hoppe and Dr. K. Pulli for the range data and Dr. C.M. Leonard for the brain 11I data.


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(a) (b)


(c) (d)


(e) (f)


Fig. 5. 1 !I ii, the dynamic open surface model to discrete points in 3D : (a) model initialization depicting
the associated control mesh, (b) model and data points, (c) & (d) intermediate stages of the fitting,
(e) final fit depicting the control mesh and (f) fitted model without control mesh.


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(a) (b)


(c) (d)


(e) (f)


Fig. 6. 1 !,II1, the dynamic closed surface model to discrete points in 3D : (a) model initialization
depicting the associated control mesh, (b) model and data points, (c) & (d) intermediate stages of
the fitting, (e) final fit depicting the control mesh and (f) fitted model without control mesh.


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(a) (b)


(c) (d)


(e) (f)


Fig. 7. (a) Range data of a bulb, (b)initialized model and the associated control polygon, (c) data and
superimposed initialized model, (d) intermediate stage of evolution, (e) fitted model and (f) fitted
model with its control polygon.


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(a) (b)


(c) (d)


(e) (f)


Fig. 8. (a) Range data of a head, (b)initialized model and the associated control i.'.1 _-.,i (c) data and
initialized model, (d) intermediate stage of evolution, (e) fitted model and (f) fitted model with its
control I'*1 -**. i


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(a) (b)


(c) (d)


(e) (f)


Fig. 9. (a) Range data of an anvil, (b)initialized model and the associated control i" .1- _- *i (c) data and
initialized model, (d) intermediate stage of evolution, (e) fitted model and (f) fitted model with its
control I,1, -_ -i


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(c) (d)


(e) (f)


Fig. 10. (a) A slice from a brain 11i 1i (b) initialized model inside the region of interest superimposed on
the slice, (c) intermediate stage of the evolving model, (d) fitted model, (e) & (f) arbitrary 3D views
of the model fitted to the cerebellum.


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(a) (b)


(C) (d)


(e) (f)


Fig. 11. (a) Data points ;.1. i li ii- the boundary of the region of interest (a caudate nucleus) on a liI
slice of human brain, (b) data points (from all the slices) in 3D, (c) data and initialized model, (d)
intermediate stage of evolution, (e) fitted model and (f) another view of the fitted model.


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