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Group Title: Department of Computer and Information Science and Engineering Technical Reports
Title: From global to local, a continuum of shape models with fractal priors
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Title: From global to local, a continuum of shape models with fractal priors
Series Title: Department of Computer and Information Science and Engineering Technical Reports
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Creator: Vemuri, Baba C.
Radisavljevic, A.
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From Global to Local, a Continuum of Shape

Models with Fractal Priors *


B. C. Vemuri1 and A. Radisavljevic2
1Department of Computer & Information Sciences
2Department of Electrical Engineering
University of Florida, Gainesville, FL 32611




Abstract
3D Shape modeling has been a very prominent part of Computer Vision over the
past decade. Several shape modeling techniques have been proposed in literature, some
are local (distributed parameter) while others are global (lumped parameter) in terms
of the parameters required to describe the shape. Hybrid models that combine both
ends of this parameter spectrum have been in vogue only recently. However, they do
not allow a smooth transition between the two extremes of this parameter spectrum.
In this paper, we introduce a new shape modeling scheme that can t,.-f, n,,'
smoothly from local to /glhal models or vice-versa. The modeling scheme utilizes a
hybrid primitive called the deformable superquadrics constructed in an orthonormal
wavelet bases. These multi-resolution bases provide the power to continuously trans-
form from local to global shape deformations and thereby allow for a continuum of
shape models -from those with local to those with global shape descriptive power
to be created. The characteristic of continuously transforming from local to global
shape deformations allows us to generate fractal surfaces of arbitrary degree that can
be useful in describing natural detail.
We embed these multi-resolution shape models in a probabilistic framework and
use them for segmenting anatomical structures in the human brain from MRI data.
A salient feature of our modeling scheme is that it can naturally allow for the incor-
poration of prior statistics of a rich variety of shapes. This stems from the fact that,
unlike other modeling schemes, in our modeling, we require relatively few parameters
to describe a large class of shapes completely.


*This work was supported in part by the University of Florida Brain Institute









1 Introduction


Modeling shapes is an important and integral part of Computer Vision as well as Computer

Graphics. In Computer Vision, modeling shapes is important for shape reconstruction and

shape recognition from sensed data, while in Computer Graphics, the application is for shape

synthesis. Many shape modeling schemes have been proposed in the literature. In Computer

Vision, the motivation for modeling shapes has been primarily driven by either shape re-

construction or shape recognition tasks. Shape reconstruction from sensed data requires a

broad geometric coverage. The models must recover detailed structure from noisy data using

only the weakest of the possible assumptions about the observed shapes. Generalized spline

models with continuity constraints are well suited for fulfilling the goals of shape reconstruc-

tion (see [16, 4]). Generalized splines are the key ingredient of the dynamic shape modeling

paradigm introduced by Terzopoulos et. al., [18]. Incorporating dynamics into shape mod-

eling with generalized splines, allows for creating realistic animation for computer graphics

applications and also for tracking moving objects in computer vision applications. With the

advent of the dynamic shape modeling paradigm, there was a flurry of research activity in

this area, with numerous application motivated modifications to the modeling primitives,

and the external forces derived from data [20, 21, 22, 8, 5]. However, the distributed na-

ture of these models -each point of the spline surface potentially contributes three spatial

degrees of freedom -makes them unsuitable for shape recognition tasks. Shape recognition

-a high-level vision task -requires that the shape models be characterized by a small set

of parameters. Lumped parameter models are thus better suited for such tasks. Some ex-

amples of lumped parameter models include, superquadrics [3, 12, 2] and the parametrically

deformable models [23, 9].

Recently, a hybrid modeling scheme dubbed, deformablee superquadrics" was introduced

by Terzopoulos et al., [19]. These models have the advantage of combining the descriptive

power of lumped and distributed parameter models and thus, simultaneously satisfying the

requirements of shape reconstruction and shape recognition. However, these hybrid models

do not exhibit a smooth transition in the number of parameters required for their description.









For example, the two extremes of the spectrum are occupied by lumped models at one end

and distributed models at the other. The former requiring, few parameters and the later

requiring a large number of parameters for their description. The modeling scheme described

in [19] involves a superposition of the lumped and distributed parameters to completely

characterize the plethora of generated shapes. Although, the hybrid models can be used to

generate both local and global shape deformations, they do not depict a smooth transition

between these types of deformations. In addition, there is no smooth transition in the number

of parameters that can capture descriptions of the range of generated shapes.

In this paper we introduce a new multi-resolution dw.,ii;'. modeling scheme that uses

an orthonormal wavelet basis [11, 7]. Our models have the unique property of being able

to ,ii..,,'/ lilscale up or down the range (from local to global) of possible deformations and

the number of parameters required to characterize them. This is achieved by adding to the

global (lumped) parameters of the deformable superquadrics, a set of wavelet coefficients

of the membrane spline at any resolution. Earlier work reported in Pentland [13] on the

use of wavelets in the context of shape modeling was limited to surface reconstruction. He

used wavelets basis to approximate the Eigen vectors of the stiffness matrix assembled in the

finite element solution to the surface reconstruction problem using the thin-plate-membrane

splines. The main idea of his work was to speed up the solution to the surface reconstruction

problem by a change of basis, from regular nodal basis to the natural basis represented by

the Eigen vectors of the stiffness matrix. Wavelets basis were used as an approximation

to the natural basis i.e., the Eigen vectors of the stiffness matrix. In the wavelet basis,

the stiffness matrix is approximately diagonal and this approximate diagonalization leads to

savings in storage space as well as computation time for the solution. In our research, the

key focus is on development of a general shape modeling scheme which when embedded in

a probabilistic framework can be utilized for easily incorporating a larger class of shapes as

prior information. In addition, we also emphasize on the efficient computation of the model

fitting process to the data.

The thin-plate-membrane regularizers were also used for a very different purpose by Szeliski

et. al., [15] namely, generation of constrained fractals which are useful in representing natu-









ral images. They derive the fractal characteristic of a membrane and thin-plate regularizers

and show that the membrane splines give raise to fractals of degree two while the thin-plate

splines yield degree four fractals. They approximate a fractal of in-between degree by using

a blend of the membrane and thin-plate models. The modeling scheme we introduce in this

paper allows us to generate fractals of in-between degree without having to resort to approx-

imations with a blend of the plate membrane models. We achieve this by modulating the

frequency characteristics of the wavelet basis with a polynomial of appropriate degree. In

order to generate stochastic fractal surfaces, we convert the energies into probability distri-

butions using ideas from statistical mechanics [10] and employ a stochastic multi-resolution

algorithm. We will present some examples of the synthesized fractal surfaces in a subsequent

section.

The rest of the paper is organized as follows: In section 2 we briefly describe the geometry

of the modeling scheme and the multi-resolution basis for expressing the geometry, followed

by a new way to augment the global shape of the deformable superquadrics primitive for

achieving the smooth transition from global to local shape descriptions. Section 3 contains

discussion on how to embed our models in an probabilistic framework and the derivation of

the mechanics of the modeling scheme. In section 4 we derive the internal energy of the prior

model in the wavelet basis and present the fractal characteristics of this prior model with

the aid of synthesized fractal surfaces. Section 5 contains a description of, the unsupervised

learning technique that we use for incorporating specific a priori knowledge into our modeling

framework and, implementation results on real MRI data followed by conclusions in section

6.


2 Multi-resolution Geometry of the Modeling Scheme


In this section, we will briefly describe the geometry and the construction of a multi-resolution

wavelet basis for the modeling scheme. To dramatically increase the library of shapes that

can be modeled and to achieve a smooth transition from global to local shape descriptions,

we will present a new method that will augment the global parameters of the deformable









superquadrics.


2.1 Geometry of the Deformable Superquadrics

The deformable superquadrics are closed surfaces in space whose intrinsic coordinates u =

(u, v) are defined on a domain Q. The positions of points on the model in an inertial frame
of reference are given by a vector-valued function x(u) = (xI(u), x2(u), x3(u))T. In a model-

centered coordinate frame, the position vector x becomes

x = c + Rp (1)

where c is the location of the center of the model, and the rotation matrix R specifies

the orientation of the model-centered coordinate frame. Hence, p(u) denotes the canonical

positions of points on the model relative to the model frame. We further express p as the sum

of a reference shape s(u) and a displacement d(u) namely p = s + d. For the parameterized

reference shape s(u) we use the superquadrics with the bending deformation (as in [2]), due

to it's attractive global shape characterization. The reference shape s is given by,

s = B(s) (2)

Where B is a bending deformation, a function of two parameters: the radius of curvature

k and angle of the bending plane a (see [2] for details). The parametric equation of a

superquadric s is given by
/aCJC1 S 2
S= a2CU6SU2 (3)
a3 Se1
Where, -r/2 < u < v/2 -7 < v < w, and S = sgn(sin w) sin wl| and C' = sgn(cos w)|cos wl .

Here, 0 < al, a2, a3 < 1 are aspect ratio parameters, (1, 62 > 0 are "squai. i. - parameters.

This defines the geometry of our underlying reference shape namely, the superquadric. We

collect all these shape parameters along with the two bending parameters into a vector q,.

We will denote c by the vector qc and use qo to denote the vector of rotational coordinates

of the model in a quaternion representation (see [19]).

An important aspect of our reference shape which differs from previous work of Terzopou-

los et al., [19] and Bajscy et al., [2], is that we have constructed a uniform tessalation of









the superquadric reference shape. Superquadric shapes exhibit anomalous behavior when
the squareness parameters 1, 62 depart from unity, causing grid lines to become unevenly
spaced leading to distortion of the deformational properties of the superimposed membrane
surface and inefficient use of computational resources. In addition, numerical instabilities
may be caused in the computation of derivatives due to the large grid spacing. A Uniform
tessalation of the superquadric can be used to overcome these problems. To achieve the
uniform tessalation, we first construct uniformly spaced grid lines in the parameter direction
u and apply the same technique to achieve uniform spacing along the parameter direction
v. Thus, we seek a function of the parameter u (same argument applies to the parameter v)
which when substituted into the equation of the superquadric i.e., equation 3, would yield
a constant differential arc length increment for a constant differential increment in u. For a
horizontal elliptical contour,
ds= (x/)2+ (yIdu (4)

Setting ds/du = constant = (circumference/27) and using (x = alcos'(f(u)), y =
a2sin-(f(u)), z = 0) results in an ordinary differential equation for f(u)

df I tan(f) (
= K (5)
du e/a# l tan(f) 41 cos(f) 2 +a sin(f) 12
f(O) = 0 as the initial condition (6)

Where K is a constant. This ordinary differential equation can be solved numerically. In
figure 1, a superquadric with its natural nonuniform tessalation is shown side by side to the
same superquadric ,.., l,,, tessellated using the method described above.
In the following sections, we show that the displacement vector function d = )Sqd is
completely determined by a set of wavelet coefficients collected into the vector qd, with
S corresponding to the discrete wavelet transform and ) containing nodal interpolation
functions. Thus, the state of the model is fully expressed by the vector q =q q q q T q .

2.2 Multi-resolution Analysis and Wavelet Bases

Multi-resolution analysis has been quite popular in computer vision research for the past
decade. Most of the research in the past has dealt with speeding up computations in solu-
























Figure 1: Tessellations of a Superquadric, (a) nonuniform (b) uniform

tions of various low-level vision problems using multi-resolution techniques [17]. With the

advent of the wavelet transform (WT), multi-resolution analysis of functions has become

more attractive due to the fact that wavelet decomposition of a function (signal) into dif-

ferent frequency components allows for studying each component at a resolution matched

to its scale. In the following, we will very briefly describe the construction of a wavelet

based multi-resolution approximation in 1D and its extension to a 2D function specifically

the membrane displacement function d(u, v).


2.2.1 Detail Function from Multi-resolution Approximations

A multi-resolution approximation of a scalar valued function d(u) e L2 (R) at a resolution j

is obtained by projecting d(u) into a vector subspace Vj C L2(1) where integer (k) translates

Pjk(u) = 2-/-j (2- u k) of a dilated scaling function 0(u) form the orthonormal basis of
Vj. The approximation dj(u) is expressed as :


d((u)= ^JlkJU}k
k


Where the set of coefficients ack = (d(u), Ojk) for all integer k corresponds to the discrete

characterization of this approximation.

An important property of multi-resolution approximations is that all the necessary infor-

mation to compute the function at a smaller resolution j + 1 is contained in the function at
























Figure 2: Division of the frequency domain for a) multi-resolution and b) wavelet represen-
tation.

resolution j. In other words the subspaces Vj satisfy the containment hierarchy,

.....V3 C V2 C V1 C VO (8)

The scaling function 0 has the effect of a low-pass filter with the cutoff frequency decreasing

proportionally to 2-j (see figure 2(a)). The key idea of the wavelet basis is to span the

difference spaces Wj+1 between the two approximations Vj and Vj+1 such that Wj+1 I Vj+i.

A wavelet decomposition therefore consists of splitting the function at resolution j into the

detail function in Wj+1 and the next lower (j+1) resolution approximation in Vj+i. This

leads directly to a scale wise, orthogonal decomposition of L2(R). To represent the function

d(u) of a finite resolution we can assume d(u) e Vo, where Vo is given by,

..... W3 ( w2 ( w1 = (9)

with a denoting the direct sum operator. The hierarchicalorthonormal basis for d(u) is given

by a family of dilations and translations of a wavelet function 0(u) : Ojk(u) = j(2-ju -

k). Thus, d(u) can be be written in this basis as,


d(u) = ;' "(u) (10)
j,cc cc

Where the discrete wavelet coefficients are computed as Sjk = Kd(u), Yjk).









As in the case of the multi-resolution approximation, wherein a low pass filtering interpre-
tation was given to the scaling function, a filtering interpretation can also be given to the
WT. At each scale j the WT corresponds to a logarithmically spaced band-pass filter (see
Figure 2(b)). Our work in this paper is primarily based on this filter bank interpretation of
the WT as discussed in Rioul et. al. [14].


2.2.2 Extension to 2-D and Fast Implementation

Extension to 2D of the multi-resolution approximation using wavelet basis can be easily
achieved using separable wavelets obtained from the products of one-dimensional wavelets
and scaling functions [11]. For the multi-resolution approximation of the membrane d(u, v)
we use the tensor products of p and y given by ( see Mallat [11]),

'"(UV) = 9(U)jk0 (v)jl
'(UV) = p(U)j (V)j, (11)


Note that, since the displacement d is a vector valued function, we have to repeat the tensor
product approximations shown above for each component of the displacement vector. The
superscript H (L) represents high (low) pass filtering in each of the directions u and v in
the parameter domain Q, and the indices k and 1 represent the integer shifts (as in the 1D
case) in u and v directions respectively. We collect the coefficients of the inner product of
the basis and the function d(u, v) i.e., 6 6&{H and 6HH into a vector qd as illustrated in
Figure 3. This displacement vector qd contains a complete hierarchical description of the
membrane surface. Note that qd has the same number of elements as the original nodal
discretization at j = 0. For our WT implementation we adopted the specific 0(u) and y(u)
functions as given in Mallat [11], which are based on the cubic spline interpolants. If we
denote the original nodal discretization at resolution j = 0 by a vector 0o then, we have,

ao =S qd Reconstruction
qd = ST ao Decomposition (12)
d(u,v) = D(u, v) ao Interpolation

where 1 represents a vector of interpolation functions (as in the 1D case, see equation 7)
and multiplication by the orthogonal matrix S corresponds to performing the WT. A fast























Figure 3: a) Organization of the displacement vector qd, b) separation of the "global" and
"local" parts of qd

implementation of the WT is achieved using Quadrature Mirror Filters (QMFs). A self
explanatory block diagram of the filterbank interpretation of the QMF is presented in figure
4 (see Mallat [11] for details). Figure 4 shows an efficient way to implement the WT, with

a computational cost of order O(N) (where N is the number of grid nodes).

Finally, vector qd is expressed in an orthonormal wavelet basis. Along with other pa-

rameters of the model it is subjected to the iterative minimization process as discussed
subsequently.

2.3 From Global to Local Shapes with Wavelet Coefficients

In figure 3, we explicitly depict the hierarchical decomposition of the displacement vector qd
in an orthonormal wavelet basis. The coefficients 6j toward the upper left corner are fewer

then those toward the bottom-right corner, i.e. the corresponding grid nodes are spaced
farther apart and the basis functions are coarser. Few parameters needed in the description

and ability to depict the global shape gives the low frequency components of the wavelet
representation a "semi-global" character. Depending on the application we can draw the

boundary at an arbitrary jo to separate the global and local part of vector qd,

qd = [qda qd]
qdg = [ ,"....,6J0 1 (13)
qdl = [j0o-i, ***, 1]



























Figure 4: a) Decomposition and b) reconstruction of a 2-dimensional array adopted from
Mallat[89]

where qd contains the top (coarser) portion of the multi-resolution pyramid. One of the

useful aspects of including qd- into the global description along with superquadric parameters

is the greater variety of shapes we can generate, such as coarse dips, bumps etc. (see figure

5). For a comparison of possible global deformations generated via tweaking of superquadric

parameters and the coarse level wavelet coefficients of the displacement function, we illustrate

the deformed shapes adjacent to one another. Figure 5 depicts the original reference shape

on the top left hand corner, global deformation resulting from tweaking a superquadric

parameter is shown on the top right hand corner. The bottom two shapes are generated via

global deformations caused by tweaking the coarse level wavelet coefficients.

When we build our prior model with mean values and covariances on q, and q, the model

assumes this enhanced shape as it's new rest state and it deforms away from this shape under

the influence of sensed data. The need for incorporating more specific information such as

this, into the prior model is especially present in difficult segmentation problems encountered

in Medical Imaging applications eg. segmentation of anatomical shapes from MR images.

We will present two such examples in this paper.






























Figure 5: a) Global shape deformations generated via tweaking global parameters of the
superquadric model, b) global deformation generated by tweaking of coarse scale wavelet
coefficients

3 Probabilistic Framework and Model Mechanics


In the last section, we have described our multi-resolution deformable superquadric model

in a wavelet basis. The multi-resolution wavelet basis provides a computationally efficient

solution to the problem of fitting the deformable superquadric model to data sets and also

provides us a way to enhance the library of shapes that can be described using a few global

parameters. In this section, we will present a way to incorporate specific prior knowledge

about the shape to be extracted from the data and then derive the mechanics of our modeling

scheme.

We cast the model fitting process in a probabilistic framework and incorporate prior dis-

tributions on the vector of geometric parameters that we are estimating. The combination

of prior and sensor models results in a Bayesian model, since the posterior distribution of the

parameters we are trying to estimate conditioned on the data can be computed using Baye's

rule. In this paper, we use the multi-resolution physically-based shape models described in

earlier sections as the prior models. We collect the statistics on the global parameters of the









model via a training process described in a subsequent section and for the local parameters,

we convert the continuous strain energy that governs the deformation of the deformable su-

perquadric model away from its natural shape into a probability distribution over expected

shapes, with lower energy shapes being the more likely. This is achieved via a technique of

statistical mechanics -conversion of energies into probabilities using Boltzmann, or Gibbs,

distribution [10]. The link between physically-based deformablee models) models and priors

is conveniently established using a Gibbs distribution of the form

1
p(q)= -ep(-E,(q)), (14)
Zp

where E,(q) is the discretized version of the internal smoothness energy of the model, and

Z, (called the partition function) is a normalizing constant. What was originally an elastic
energy restoring a model towards a rest state now becomes a probability distribution over

expected shapes, with lower energy shapes being more likely. To complete the formulation

of the estimation problem, we combine this prior model with a simple sensor model based

on linear measurements with Gaussian noise

1
p(D/q) = -ep(-ED(D, q)), (15)
ZD

where ED(D, q) can be either an edge-based potential energy synthesized from the MR

images or the potential energy in the springs attached from 3D data points to the surface

of the deformable superquadric constraining it to conform to the observed data. Combining

the prior and sensor models (from equations 14 and 15 respectively) using Baye's rule, we

obtain the posterior distribution

p(D/q)p(q) 1
p(q/D) -ep(D= (-E(q)), (16)
p(D) Z

where


E(q) = E(q) + ED(D, q). (17)

Computing the Maximum A Posteriori (MAP) estimate, i.e., the value of q that maximizes

the conditional probability p(q/D), provides the same result as finding the minimum energy









configuration of the physically-based model. However, the advantage of the probabilistic
framework being that, it allows for explicitly incorporating prior model and sensor model
characteristics in addition to providing an uncertainty in the estimates.
The internal strain energy Ep of the model in our case is given by a quadratic form,
Ep = 1/2[(q q) TK(q q)]. Where, q is the rest state of the model and the matrix K]
corresponds to the stiffness of deformation. We will discuss the internal energy Ep further in
a subsequent section. The two types of sensor model energies for ED(D, q) discussed earlier
are expressed as

f i /3(D x(q, u)2 spring energy
ED(D, q)= /3 |P(x(q, u))| image potential (

In the above equations ui determines the closest point x to the given measured point Di.
We adjust the locations of these grid coordinates as the shape of the model evolves (see [19],
for more on migrating point of influence). The magnitude of parameter 3 is related to the
uncertainty of sensor measurements. P(x(q,u)) denotes an edge based potential derived
from the MR image data. Maximization of of the posteriori amounts to minimizing the
total energy E = Ep + ED for which we employ a gradient descent method that amounts to
iteratively solving,
qk+l = qk AE(q) (19)
Oq
For the derivative of the energy on the right hand side in the above equation, we use the
chain rule to get a= Oa ax where the contribution of each parameter group is

ax ac a(Rp) Ras RO(4ISqd)
q 9qc,' sq0 9q,' 09qd
= [I B RJ iS] (21)

= L (22)

Here, we adopt the notation as in [19]. Partial derivatives of the prior model and data
energies defined above can be now written as

=E)
OE = K (q tj) (23)
aq









and
OED Eq /(Di x(q, u1)L spring energy
Oq -q = V |P(x(q, u)) L image potential (24)
By substituting equations 23 and 24 into the equation for gradient descent 19 we obtain

an expression for iterative updating of the state vector q

(q(k+) (k) +A (fk) K(q(k) ) (25)


where A is a diagonal matrix containing step sizes and consequently effecting the speed

of convergence. Close similarity of this evolution equation to the one derived in [19] reveals

that equation 25 can be modified slightly and effectively used to generate motion sequences

corresponding to first order dynamics Cql + K(q q) = fq. In addition to producing the

statistically most likely state q, this framework provides the uncertainty in the estimate

via the associated covariance matrices. In our work, we employ the above gradient descent

procedure in a multi-resolution framework for computational efficiency as well as alleviation

from entrapment in local minima.


4 Prior Model in Wavelet Basis

In the previous section we described a probabilistic framework for embedding our model. We

now derive the explicit form of the internal energy of the model in the wavelet basis. This

energy is then converted to a Gibbsian probability distribution using the technique discussed

in section 3.

4.1 Internal Energy of the Prior Model

In addition to generic smoothness assumptions, a prior model should incorporate the infor-

mation about the possible shapes that an observed object can assume. This can be very

important in applications involving segmentation of known natural shapes from "unfriendly"

(low contrast and noisy) images. We express the probability of a given configuration q

through a Gibbs distribution (see section 3), which assigns a low probability to configura-

tions with large energy Ei and vice versa. The configuration with Ei = 0 represents the








rest state of the model. The internal energy of the membrane spline (which augments the
reference superquadric shape s of the model), in the continuous form is given by,

Ea = / wod2 + w+ a) du (26)

With no additional prior information about the shapes being modeled, the underlying
global shape of the model namely, the superquadric, can be interpreted as the mean value
or the rest state of the membrane surface -expressed by the displacement function d. If we
rewrite the deformation energy in terms of vector p = s + d we have an equivalent expression
to equation 26,
/((O 8(p s) 2 + p S)
Ea(p S. -+) du (27)

Where the role of the global (reference) shape s as the rest state becomes apparent. We
extend our underlying global shape s with a low resolution mean displacement function
d = dt0 corresponding to parameters qdg as defined in equation 13 and shown in figure 3.
This is done by altering the deformation energy equation 26 to,

E = / wo(d d)2 + w (d d) + ((d d) ) du (28)
.a1 u av
The discrete version of this internal strain energy can be written as,
1
ET = -(qd qd)T Kd (qd qd) (29)
2
Where, Kd is the stiffness matrix that determines the elastic properties of the model and
qT = [qdg, 0,...,0] incorporates the statistics of the possible shapes that can be encoun-
tered in the segmentation problem. These statistics are gathered through training process
described in section 5. It is evident that the number of parameters in qd, is low, i.e. the
representation of the model remains compact and thus making the training process feasible.
We include other global shape parameters of the model in our prior statistics by extending
the above energy as follows,

E = q(q q-)T C (q, -q) +

(qb -q)T b b 1 (qb q) +

2(qd qd)T Kd (qd qd) (30)









Where s denotes the superquadric shape without the bending while, Cs represent the co-

variance matrices which are accumulated in the iterative training phase. In section 4.2 we
show the derivation of stiffness matrix K in the wavelet basis and comment on its structure.

4.2 The Stiffness Matrix in Wavelet Basis

The stiffness matrix for the membrane operator is a very large, sparse and banded matrix in

the ordinary nodal basis. By expressing this in an orthonormal wavelet basis, we can achieve

near diagonalization of this matrix. We will show how the filter bank interpretation of WT

(see figure 2) relates directly to the diagonal entries of the stiffness matrix K. This approach
in turn gives us a straight-forward and inexpensive way to compute K.

The standard way to obtain stiffness matrix in wavelet basis is essentially a similarity

transformation. By combining equations 12, 28 and 29 we get,

KO = woST (f 4 du) S (31)

K1 = wiS Q/< l) du) S (32)

K = K + K1 (33)

Here Ko controls the local magnitude and K1 the local variation of the (membrane) displace-

ment function. Due to orthogonality of the interpolation functions in ) and orthogonality

of S, equation 31 simplifies to K = wol. However, the part K1 of the stiffness matrix in

equation 33 needs to be computed by a similarity transformation K1 = STjI S, where K1

represents the membrane stiffness matrix in nodal basis (see [13]). The drawback of comput-

ing K1 using this similarity transformation is the large computational cost incurred (O(N2),

where N is the number of nodal variables) which can be avoided by using the property that

differential operators in wavelet basis remain sparse (see [1]) and have strong diagonal dom-

inance (see [13]). This allows us to compute the diagonal elements of the stiffness matrix

directly by taking inner products of the corresponding finite element basis functions (an

O(N) computational cost),


K3k = W1 2 Tk() 2d (34)
--oo

























Figure 6: (a) Scaling of the diagonal entries in the stiffness matrix K of the membrane
model with spectral density, a-2 and an arbitrary density w-~, (b) Increasing energy with
increasing resolution

which is done in frequency domain by employing Parseval's theorem [6]. Extension to 2-

D case is straightforward and achieved by using the tensor product of the basis functions.

The factor jw results from the Fourier transform of the first derivatives representing the

membrane regularizer. Note that both K and K1 have diagonal structure and that only the

relative magnitude of their diagonal entries determine the smoothing effects of each matrix.

The filter bank interpretation of WT (Figure 2) explains how coefficients corresponding

to lower resolutions j contribute less energy than those corresponding to higher resolutions.

Figure 6 illustrates two aspects of the modeling, (a) the membrane displacement function

which has a spectral density characterized by w-2 and is a fractal process of degree two,

can be used to modulate the ideal bandpass characteristics of the wavelet basis for creating

the appropriate stiffness matrix. In addition, one can synthesize fractals of arbitrary degree

by using a polynomial function w-~ to modulate the bandpass characteristic of the wavelet

basis for creation of the appropriate stiffness matrix; (b) The concept of increasing energy

contribution with increasing resolution via a 1D example. For instance, if we assume an

ideal band-pass characteristic for Tj(w) between wj and 2wj we would have the following

relationship between elements of K at j and (j + 1) resolutions.


Kik = w Ijw l2d = 8I1(j+i)k (35)
W]A



























Figure 7: Diagonal entries of K as an elevation map, for a (32X32) grid computed via a
similarity transformation

Therefore, for the membrane regularizer, we notice that the energy at a resolution (j + 1) is

eight times that at the next lower resolution j.

For a verification of the above discussed sclaing behaviour, we computed the diagonal

entries of the K matrix for a (32 X 32) grid of nodes using a similarity transformation

technique discussed earlier. Figure 7 depicts the diagonal entries as an elevation map,

where the elevation corresponds to the magnitute of the entries. The entries are organized

as in a standard representation of images in wavelet basis (see figure 3 for the standard

organization).


4.3 Prior Model of General Fractal Degree

Splines and stocahstic fractals are complementary techniques for modeling free-form shapes.

Splines are well suited for modeling smooth man made objects and are easily controlled.

Fractals on the other hand are well suited for modeling irregular shapes and are difficult to

constrain. Szeliski et. al. [15] describe a way to constrain fractals by establishing a formal

connection between splines and fractals using Fourier analysis of energy minimizing splines.

Our prior model corresponding to the membrane spline is a Markov random field with a

Gibbsian distribution. Its spectrum can be determined using Fourier analysis techniques. In
























Figure 8: Surface fitting result a) MAP Estimate and b) Representative sample from the a
posteriori distribution, corresponding to a fractal surface (membrane model)

their work on constrained fractals Szeliski et. al. [15] show how prior models corresponding

to a membrane and thin-plate regularizers posses a spectrum w-2 and -4 respectively, which

is fractal i.e., it is self-affine over scale. These spectra characterize special cases of fractals

of general form given by,


1
S(w) oc (36)

In [15] fractals of an inbetween degree 2 < 3 < 4 were approximated by blending the

membrane and thin-plate terms. In contrast, our multi-resolution wavelet based modeling

scheme discussed earlier allows us to construct fractals of any degree directly by utilizing

the natural scale space characteristics of wavelet basis. Elements of K governing the prior

distribution are computed using either equation 34 or 35 with w2 replaced by 2A. Thus,

yielding a stiffness matrix corresponding to regularizer whose spectral distribution is of order

3. A random sample from the corresponding Gibbsian distribution of this prior model will

be a fractal surface of degree 3.

Figure 8 shows a surface fitting result to 3D point data set with (a) MAP estimate and (b)

representative sample from the Gibbs distribution p(q|D) reflecting the underlying fractal

prior. Fractal surfaces are very useful for modeling natural detail. In our application of

Medical Imaging, our goal is to realistically model the shape of the anatomical structures
































Figure 9: Supervised learning to incorporate information into the prior model

found in the human body, specifically the brain. Experiments toward achieving this goal are

currently underway.


5 Supervised Learning and Implementation Results


In our work, prior knowledge about the object shapes that are to be segmented from the

MRI data is collected via a supervised learning technique. A training sequence of several

iterations provides samples of correctly segmented shapes for specific classes of objects -in

our experiments a i/i, i, and the hippocampus from the human brain. Parameter sets q(i)

corresponding to these shapes represent the training sequence for the global part (q, + qdq)

of our prior model (see figure 9). With each iteration i we improve estimates of the mean

vector q and covariance matrix C through a simple equation for accumulating measurements

as indicated in the Figure 9. Since we update only the statistics of the global parameters

of the model, the local deformations remains constrained only by smoothness assumptions

captured in the stiffness matrix K (refer to equation 30). Therefore, most of the flexibility of









the locally deformable surface remains unchanged. We recommend human visual supervision

of the surface fitting in the training stage to assure that only good segmentation results enter

the statistics of the prior model. An extension to multi-class case is straightforward, and

can be achieved by defining multiple prior distributions pj(q) for each class j.

This method clearly demonstrates the useful aspect of global models which offer a compact

description of the overall features of the object. Deformable superquadrics with the global

shape extended by some offset deformation (as presented in previous sections) allow a much

wider range of shapes to be representable in the prior model, all at the expense of few

additional parameters. This makes our model very attractive for shape recognition as well

as shape reconstruction tasks.

We now present two examples depicting segmentation of a gyrus and a hippocampus from

MRI data. The data sets consisted of 64 slices each, with each slice of 1.25mm thickness. We

applied a simple 3D edge detection technique to create a gradient force field which was used

to attract the inialized model toward the edges of the gyrus and the hippocampus in the

first and second examples respectively. Figure 10, illustrates our results displayed with slices

of the reconstructed shape superimposed on the corresponding slices from the original data.

The region of interest has been boxed in the original image and our results are displayed

only for this region of interest. Left to right, the images are, a slice from the original data,

a slice from the initialized 3D model, the corresponding slice from the final segmentation.

Further experimentation with our modeling scheme is currently under way and we expect

to better our results by using more sophisticated 3D edge detectors [5] for defining the

external forces to draw the model into the desired shape.


6 Conclusion


In this paper, we introduced a new shape modeling scheme that can transform ,i..,... I,/

from local to global models or vice-versa. The modeling scheme utilizes the hybrid primitive

called the deformable superquadric constructed in an orthonormal wavelet bases. These bases

provide the power to continuously transform from local to global shape deformations and

































Figure 10: Segmentation results for (a) a gyrus and (b) a hippocampus from MRI data;
Figure shows slices from 3D, of data, of the initialization, and the reconstruction respectively.

thereby allow for a continuum of shape models -from those with local to those with global

shape descriptive power -to be created.

When embedded in a probabilistic framework, our modeling scheme allows us to incor-

porate more detailed prior information than when using lumped parameter models, while

keeping the number of parameters required to describe the shape low. The same level of de-

tail may also be incorporated by using some existing modeling techniques [9, 19] but, at the

expense of increasing the number of parameters tremendously. From a computational point

of view, our modeling scheme is very attractive for segmentation purposes as multi-resolution

relaxation is a natural by product of implementing the gradient relaxation method in the

wavelet basis. Also, entrapment in local minima during the minimization of the nonlinear

energy function can be alleviated. Our future efforts will be directed toward testing with

more MRI data and using better 3D edge detectors to define external forces.









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