INTELLIGENT TOPOLOGY PRESERVING GEOMETRIC

DEFORMABLE MODEL

Renato Dedi

´

c

D

´

epartement de Math

´

ematiques, Universit

´

e de Sherbrooke, 2500, boul. de l’Universit

´

e J1K 2R1, Sherbrooke, Canada

Madjid Allili

Department of Mathematics, Bishop’s University, 2600 College St. J1M 0C8, Lennoxville, Canada

Keywords:

Topology Preserving GDM, Level Sets, Deformable Models, Texture, Edge Detection.

Abstract:

Geometric deformable models (GDM) using the level sets method provide a very efﬁcient framework for im-

age segmentation. However, the segmentation results provided by these models are dependent on the contour

initialization. Moreover, sometimes it is necessary to prevent the contours from splitting and merging in order

to preserve topology. In this work, we propose a new method that can detect the correct boundary information

of segmented objects while preserving topology when needed. We adapt the stoping function g in a way that

allows us to control the contours’ topology. By analyzing the region where the edges of the contours are close

we decide if the contours should merge, split or remain the way they are. This new formulation maintains the

advantages of standard (GDM). Moreover,the topology-preserving constraint is enforced efﬁciently therefore,

the new algorithm is only slightly computationally slower over standard (GDM).

1 INTRODUCTION

The class of geometric deformable models(GDM) in-

troduced in (Caselles et al., 1993; Caselles et al.,

1997; Malladi et al., 1995) are deforming contours

(curves and surfaces) represented implicitly as level

sets of some higher dimensional scalar function.

This level sets representation allows these models

to have numerous advantages such as providing efﬁ-

cient computational schemes, automatically handling

topology changes of the evolving contours and sim-

ple implementation. These numerous advantages can

be used proﬁtably to provide a very efﬁcient frame-

work for image segmentation, edge detection, shape

modeling, and visual tracking. (GDM) level sets for-

mulation can automatically handle topology changes

and usually it is a desired property. However, topolog-

ical ﬂexibility is not always desired especially, when

a particular object is sought and its number of compo-

nents and the homology of each component is known.

In past, there have been several postprocessing meth-

ods reported to correct the topology of a cortical seg-

mentation that has the wrong topology (Shattuck and

Leahy, 2000; B. Fischl and Dale, 2001; X. Han and

Prince, 2001; X. Han and Prince, 2003; Alexandrov

and Santosa, 2005). In this and similar applications

the topology ﬂexibility of geometric deformable mod-

els is considered to be a liability rather than an advan-

tage (X. Han and Prince, 2001). Although ”snakes”

introduced by (M. Kass and Terzopoulos, 1987) do

preserve topology they do not give us the ﬂexibility to

change the topology if needed.

In this paper, we develop an intelligent topology-

preserving GDM (TPGDM) that can guarantee that

the ﬁnal contour has exactly the same topology as the

initial one but also it can let the contours merge or

split when judged appropriate.

This paper is organized as follows. In Section 2, we

brieﬂy introduce the geometric deformable models.

In Section 4, we explain the algorithm for contour ini-

tialization. In Section 5 we explain the new TPLSM.

An experimental result is also presented. A brief con-

clusion is given in Section 7.

2 GEOMETRIC DEFORMABLE

MODELS

Geometric models for active contours have brought

tremendous impact to classical problems in imagery

322

Dedi

´

c R. and Allili M. (2007).

INTELLIGENT TOPOLOGY PRESERVING GEOMETRIC DEFORMABLE MODEL.

In Proceedings of the Second International Conference on Computer Vision Theory and Applications - IFP/IA, pages 322-327

Copyright

c

SciTePress

such as providing ways to devise efﬁcient com-

putational algorithms for automatic segmentation.

This is achieved by using the level set methods,

which allow handling automatic changes in topol-

ogy while providing a framework for very fast nu-

merical schemes.These models are based on the the-

ory of curve evolution and geometric ﬂows. The

curve/surface is propagating (deforming) by an im-

plicit velocity that contains two terms, one related to

the regularity of the deforming shape and the other at-

tracting it to the boundary. The model is given by a

geometric ﬂow(PDE), based on mean curvature mo-

tion, therefore it’s completely intrinsic. When imple-

mented using the level set based numerical algorithm,

the model handles topology changes automatically.

The geometric model proposed by Caselles et al

(Caselles et al., 1993) is based on the mean curvature

motion equation which describes the propagation of

the level set function following the normal direction

with speed depending on the mean curvature. Let u

be a level set function u : R

2

×[0, +∞) → R and curve

C is a level set of u, such that C = {x ∈ R

2

: u(x, t) =

r}, r ∈ R. The geometric model is deﬁned as follows:

∂u

∂t

=| ∇u | (div(g(I)

∇u

| ∇u |

)) (1)

u(x, 0) = u

0

(x) (2)

where u

0

is the initialized curve. A similar formula-

tion called the geodesic model gives:

∂u

∂t

= g(I)(c + k) | ∇u | +∇g · ∇u (3)

where g(I) is the stopping function,

g(I) =

1

1+ | ∇

ˆ

I |

2

which will stop the propagation when the evolving

front reaches the desired position, the boundary de-

tected.

ˆ

I is a convolved image that ensures the mo-

tion of C is less affected by the noise in the image.

k is the mean curvature. For the added constant term

c, we can think cg(I) | ∇u | as an extra speed in the

geodesic problem to increase the speed of the conver-

gence. The gradient term | ∇u | controls what happens

at the interior and exterior of the interface. ∇g · ∇u

denotes the projection of an attractive force vector on

the normal to the moving interface. This term allows

to accurately track boundaries with high variation in

their gradient, including boundaries with small gaps.

There are many algorithms for numerical implemen-

tation of GDM using level sets. Narrow band method

and fast marching method are two simple, computa-

tionally fast and widely used algorithms. Instead of

computing the evolution of all the level sets, which

means all the grid points, narrow band method just

updates a small set of points in the neighborhood of

the zero level set for each iteration.

However, the results of this model depend on the po-

sition of the initialized curve/surface. Different initial

positions may lead to totally different result contours.

We will discuss in detail and show some examples in

section 6.

3 THE AVERAGE SQUARED

GRADIENT

One of the measures for locally characterizing the im-

age used in (F

¨

orstner, 1994) is the average squared

gradient deﬁned as follows: with the gradient ∇g =

(g

x

, g

y

)

T

we obtain the squared gradient Γg as dyadic

product

Γg = ∇g∇g

T

g

2

x

g

x

g

y

g

y

g

x

g

2

y

(4)

The Gaussian function with standard deviation σ is

denoted by G

σ

(x, y) = G

σ

(x) ∗ G

σ

(y). This yields the

average squared gradient image

Γ

σ

g(x, y) = G

σ

∗ Γg = Γg(u, v)G

σ

(x − u, y − v)dxdy.

(5)

The three elements of Γ

σ

g(x, y) can be derived by

three convolutions.

Γ

σ

g(x, y) =

G

σ

∗ g

xx

G

σ

∗ g

xy

G

σ

∗ g

yx

G

σ

∗ g

yy

Γ

σ

g(x, y) can be diagonalised by rotation of the coor-

dinate axes and it gives Γ

σ

g = T Λ

g

T

T

= λ

1

(g)t

1

t

T

1

+

λ

2

(g)t

2

t

T

2

.

First, the trace h = trΓ

σ

g = λ

1

(g)+λ

2

(g) =

k

∇g

k

2

=

σ

2

g

x

+ σ

2

g

y

gives the total energy of the image function

or edge busyness at (x, y). We can use h = trΓ

σ

g for

measuring the homogeneity of segment-type features.

Second, the ration v =

λ

1

λ

2

of the eigenvalues gives

us the information about the orientation or isotropy.

For example, if λ

2

= 0, we have anisotropic texture

of straight general edges with arbitrary cross-section.

Third, the largest eigenvalue can give us an estimate

for the local gradient of the texture or the edge. Due

to the squaring, the phase information is lost (Kass

and Witkin, 1987) but, the variance of the orientation

is proportional to

λ

2

(λ

1

−λ

2

)

, giving us an additional in-

terpretation and showing that if λ

1

≈ λ

2

then the vari-

ance of orientation is large.

Therefore, if λ

1

and λ

2

are eigenvalues of Γg and λ

1

≥

λ

2

then: (1) if λ

1

is large compared to λ

2

, the local

neighborhood possesses a dominant orientation, (2) if

INTELLIGENT TOPOLOGY PRESERVING GEOMETRIC DEFORMABLE MODEL

323