A MULTIPHASE ACTIVE CONTOUR MODEL WITH DYNAMIC

MEDIAL AXIS CONSTRAINT FOR MEDICAL IMAGE

SEGMENTATION

Yan Zhang and Bogdan J. Matuszewski

Applied Digital Signal and Image Processing Research Centre, School of Computing Engineering and Physical Sciences

University of Central Lancashire, Preston PR1 2HE, U.K.

Keywords:

Medical image segmentation, Active contour, Shape constraint, Level-set, Multiphase segmentation.

Abstract:

A level-set based multiphase active contour model is proposed for medical image segmentation in this paper.

The proposed method allows multiple objects with very different features to be jointly segmented by simul-

taneously evolving multiple active contours, each responsible for the segmentation of a single object. In this

model, the forces exerted on each active contour mainly consists of two components. The ﬁrst component

makes use of boundary as well as regional information present in the input images. The second component is

used to impose the so-called medial axis constraint, which is related to the force induced by interaction of mul-

tiple active contours. Experimental results on real medical images are also presented to show that the proposed

method has good performances on topology preservation of multiple contours, as well as joint segmentation

of similar objects in multiple images.

1 INTRODUCTION

Segmentation is a basic yet important problem in

medical image processing, which is widely used in

medical applications such as surgicalplanning, abnor-

mality detection and treatment progress monitoring.

Originally proposed in (Kass et al., 1988) as a tool

for image segmentation, active contour models have

attracted extensive research in the past two decades.

The basic idea of the active contour is to iteratively

evolve an initial curve towards the boundaries of the

target objects driven by the combination of internal

forces determined by the geometry of the evolving

curve and the external forces induced from the image.

Image segmentation method using active contour

is usually based on minimizing a functional which is

so deﬁned that for curves close to the target bound-

aries it has small values. To solve the functional min-

imization problem, a corresponding partial differen-

tial equation (PDE) can be constructed as the Gateaux

derivative gradient ﬂow to steer the evolution of the

active contour.

Either explicit or implicit method can be applied

to numerically approximate a curve evolution PDE.

For explicit methods, an active contour can be rep-

resented in parametric form such as linear and cubic

B-splines (Precioso et al., 2005). In that case, a ﬁnite

number of points are sampled on the active contour

and move according to the calculated forces, thereby

causing the evolution of the entire contour.

For implicit (or level set) methods, an active con-

tour is embedded as a constant level set (typically

level zero) in a function deﬁned in a higher dimen-

sional space known as embedding function or level

set function. The evolution of the active contour is

carried out implicitly by the evolution of its embed-

ding function. Thanks to its inherent capability to

handle topological changes and straightforward ex-

tensibility to cope with high dimensional data, since

the pioneering work of (Malladi et al., 1995), level set

based segmentation has gained considerable research

attention and has prompted the development of a large

amount of models. Among these, two models, namely

geodesic active contour model (Caselles et al., 1997)

and Chan-Vese model (Chan and Vese, 2001), stand

out respectively as the paradigms for boundary-based

and region-based segmentation.

Most of the level-set algorithms in the aforemen-

tioned literature are focused on designing evolution

functions for a single level set. According to these

methods, although a single level set can embed mul-

tiple separate active contours, each active contour is

516

Zhang Y. and J. Matuszewski B. (2010).

A MULTIPHASE ACTIVE CONTOUR MODEL WITH DYNAMIC MEDIAL AXIS CONSTRAINT FOR MEDICAL IMAGE SEGMENTATION.

In Proceedings of the International Conference on Computer Vision Theory and Applications, pages 516-521

DOI: 10.5220/0002917005160521

Copyright

c

SciTePress

driven independently by the same evolution rule. To

simultaneously segment multiple objects with very

different features, multiphase methods have been pro-

posed to use more than one level set functions.

In (Brox and Weickert, 2006; Lankton and Tan-

nenbaum, 2008), the evolution rule for each level

set function consists of not only the traditional terms

derived from speciﬁc functionals, but also the extra

terms imposed by the proximity constraint. The prox-

imity constraint ensures that each image pixel belongs

to one and only one segmented region. To take the

proximity constraint into account, these methods em-

ploy extra terms based on the concept of region com-

petition, attempting to classify pixels on the region

boundaries only to the most probable regions they can

belong to.

In (Vese and Chan, 2002), the classic Chan-Vese

model is extended from object/background segmenta-

tion to multiple region segmentation. In this method,

different regions are binary coded by the signs of mul-

tiple level set functions so that the proximity con-

straint can be satisﬁed in an elegant and natural way.

In (Cremers et al., 2006), a quite different multi-

phase method was proposed, where auxiliary labeling

level set functions are introduced to dynamically di-

vide an image into multiple regions which are labeled

in a similar fashion as in (Vese and Chan, 2002) to

keep the proximity constraint. The uniqueness of the

method lies in that the labeled regions are not used

directly as the segmentation result but used to iden-

tify different regions so that different evolution rules

can be deﬁned in differently labeled regions by using

a single “segmenting” level set function.

Based on the authors’ two earlier papers (Zhang

et al., 2008; Zhang and Matuszewski, 2009), we

propose a level-set based multiphase active contour

model dedicated to a different type of constraint —

the topology constraint. The model consists of two

major components. The ﬁrst component, making use

of both boundary and regional information derived

from the input image, describes how each active con-

tour evolves independently. The second component

takes the interaction of multiple active contours into

account by using inter-object medial axes.

The rest of the paper is organized as follows. Sec-

tion 2 discusses the theory of the method. Section 3

shows results of applying the method on medical im-

ages, whereas the conclusions are drawn in Section 4.

2 THEORY

2.1 Hybrid Active Contour Model

First consider the case of a single active contour. Let

C denote an active contour and x denote a point in

the image domain Ω. Then, as illustrated in Fig-

ure 1, the level set function φ(x) can be deﬁned to

have the following properties: (1) C = {x : φ(x) = 0};

(2) φ(x) > 0 for x inside the contour and φ(x) < 0

for x outside. The normal of the active contour

~

N is

deﬁned as the unit direction expanding the contour.

~

N

−

~

N

C

φ(x) < 0

φ(x) > 0

Figure 1: An active contour and its level set function.

The proposed functional to be minimized can be

written as

E(φ(x)) = −

Z

Ω

P(I(x))H(φ(x))dx

+ α

Z

Ω

g(|∇I(x)|)|∇H(φ(x))|dx (1)

where: P(x) and g(x) are the regional and boundary

mapping functions related to the input image I(x),

H(x) represents the Heaviside function which has

value 1 when x ≥ 0 and 0 otherwise, and α is a scalar

factor used to balance the two terms. The ﬁrst term of

the functional is the regional term, wherein the func-

tion P(x) is designed to map image intensities, ex-

pected to be typical for the object, to positive values

with all other image intensities being mapped to neg-

ative values. The selection of the regional mapping

function P(x) is image dependent but ﬂexible. The

second term is the classical geodesic boundary term

as proposed in (Caselles et al., 1997). The bound-

ary mapping function g(x) is often chosen to be an

image edge indicator function which is a nonnegative

decreasing function of the image gradient.

By deriving the Gateaux derivative of the pro-

posed functional, the implicit PDE, describing the

evolution of active contour that minimizes the func-

tional, can be expressed as

A MULTIPHASE ACTIVE CONTOUR MODEL WITH DYNAMIC MEDIAL AXIS CONSTRAINT FOR MEDICAL

IMAGE SEGMENTATION

517

C

background

object

Figure 2: Regional term and its ﬂow.

∂φ(x)

∂t

= P(I(x))|∇φ(x)|

+ α· ∇ ·

g(|∇I(x)|)

∇φ(x)

|∇φ(x)|

|∇φ(x)| (2)

The corresponding explicit PDE can be written as

∂C

∂t

= P(I(C)) ·

~

N

+α

g(|∇I(C)|)κ− < ∇g(|∇I(C)|),

~

N >

·

~

N (3)

with κ representing the curvature of C and < ·, · >

denotes the inner product of two vectors. A syn-

thetic P(x), where object pixels are mapped to 1 and

background pixels are mapped to -1, is shown in Fig-

ure 2 in order to illustrate the geometric interpretation

of the regional term. Superimposed on the image is

an active contour with arrows showing the motion of

the contour according to the regional term P(x)

~

N in

Equ. (3). It is clear that, in this case, the regional

term can always steer the active contour to the ob-

ject boundary (where the minimization of the regional

term in Equ. (1) is obtained) as long as the initial con-

tour has common interior area with the object. This

illustrates the robustness of the regional term which

can hardly be achieved by the boundary term alone

due to its narrow effective range and high sensitivity

to image noise. The value of the boundary term lies in

the fact that it can normally lead to more accurate seg-

mentation results when relatively strong boundaries

are presented in images. In case when boundary in-

formation is weak or ambiguous, the boundary map-

ping function can be chosen as g(x) = 1. Then the

boundary term in Equ. (3) is simpliﬁed to a curvature

ﬂow term κ

~

N whose role is simply to smooth curve.

2.2 Multiphase Framework

In the proposed model, to jointly segment multi-

ple objects, multiple active contours, each associated

with a single object, need to be applied. If each active

contour evolves independently according to the evo-

lution rule in Equ. (3), a set of PDEs can be written

as

∂C

k

∂t

= V

k

(C

k

) ·

~

N, k = 1,...,n (4)

with n denoting the number of active contours and C

k

denoting the k

th

active contour. V

k

represents the ve-

locity along the normal direction of C

k

, which can be

written as

V

k

(x) = P

k

(I(x))

+α

k

g

k

(|∇I(x)|)κ− < ∇g

k

(|∇I(x)|),

~

N >

(5)

where the selections of P

k

(x), g

k

(x) and α

k

depend on

the corresponding object to be segmented.

In order to make active contours interact with

each other so that they can evolve in a more control-

lable way, a constraining component in the form of

D

k

(C

k

;C) is introduced into Equ. (4)

∂C

k

∂t

= V

k

(C

k

) ·

~

N + β

k

D

k

(C

k

;C) ·

~

N, k = 1,...,n

(6)

where C represent the entire set of active contours,

i.e, C = {C

1

,.. .C

n

}. In our study, medial axes are

found to be an elegant way to constrain the evolution

of multiple curves. The following two subsections de-

scribe how the medial axis can be related to the con-

straining component to impose topology and dynamic

shape constraints.

2.2.1 Topology Constraint

In medical images, topology of objects to be seg-

mented is often known and can be used as prior in-

formation. By using the prior, a set of initial contours

satisfying the topology can be established. The topol-

ogy constraint requires that the initial topology of the

contours should be preserved in the process of curve

evolution.

C

1

C

2

M

(a) Exclusion

C

1

C

2

M

(b) Inclusion

Figure 3: Two fundamental topological relationship studied

in this paper.

For any pair of curves, there are two fundamental

topological structures studied in this paper, namely,

exclusion and inclusion. The exclusion structure is il-

lustrated in Figure 3(a), where the two curves C

1

and

VISAPP 2010 - International Conference on Computer Vision Theory and Applications

518

C

1

C

1

C

1

C

2

C

2

C

2

C

3

C

3

C

3

M

1

M

2

M

3

Figure 4: An example of topology of three curves and their

associated medial axes.

C

2

are disjoint, i.e., Ω

1

∩Ω

2

=

/

0 with Ω

k

denoting the

interior area ofC

k

throughout the paper. The inclusion

structure is illustrated in Figure 3(b), where one curve

contains the other, i.e., Ω

1

⊂ Ω

2

or Ω

2

⊂ Ω

1

. More

complicated topology can be built based on these two

fundamental structures. A example of three curve

conﬁguration is shown in Figure 4. For the shown

conﬁguration, the topological constraints can be writ-

ten as follows: Ω

1

∩ Ω

2

=

/

0, Ω

1

⊂ Ω

3

and Ω

2

⊂ Ω

3

.

For each evolving curveC

k

, a medial axis denoted

as M

k

is associated with it. The medial axis, depen-

dent on multiple curves, is deﬁned as

M

k

(C) =

x : d(x;C

k

) = min

j6=k

d(x;C

j

)

(7)

with d(x;C) representing the Euclidean distance be-

tween the point x and the curve C. The medial axes

are shown as the dash curves in Figure 3 and Figure 4.

Note that the medial axes are identical if there are only

two curves. The details of an efﬁcient way to calculate

medial axes can be found in (Zhang and Matuszewski,

2009).

A medial axis divides the image domain into two

regions — the region that only contains the associ-

ated active contour and the region that contains all

the other active contours. The topological structure

of multiple active contours is guaranteed to be pre-

served if none of the active contours ‘leak’ through

their corresponding medial axes to the other sides of

the regions. To prevent the leakage from happening,

forces induced from the medial axes can be exerted

on the corresponding active contours. Hence, the con-

straining term in Equ. (6) can be deﬁned as

D

k

(x;C) = f

k

(sdf(x;M

k

(C))) (8)

where sdf(x;M

k

) represents the signed distance func-

tion which equals to d(x;M

k

) for the point x belong-

ing to the region that contains C

k

and −d(x;M

k

) oth-

erwise. f

k

(x) in Equ. (8) is the so-called distance-

to-force transfer function which maps the signed dis-

tance to medial axis to the force exerted on the as-

sociated active contour. An example of the transfer

function is f

k

(x) = x − γ with the positive distance

threshold γ, which indicates that the force exerted on

the associated active contour becomes more and more

repulsive (negative) as the active contour approaches

to its medial axis from its own side.

2.2.2 Dynamic Shape Constraint

Medial axis can also be used as a dynamic shape con-

straint for joint segmentation of multiple objects with

similar shapes in different images.

C

1

C

2

Figure 5: Basic idea of using medial axis as a dynamic

shape constraint.

The basic idea of dynamic shape constraint is il-

lustrated in Figure 5, where C

1

and C

2

are two active

contours taken from two different images and put into

the same ﬁgure for demonstration purpose. In this

case, the medial axis of the two curves, shown as the

dash curve in the ﬁgure, is deﬁned as

M(C) = {x : d(x;C

1

) = d(x;C

2

) and x 6∈ Ω

1

∩ Ω

2

}

(9)

Consequently, the constraining term in Equ. (6) can

be deﬁned as

D

k

(x;C) = sdf(x;M(C)) (10)

where sdf(x;M) represents the signed distance func-

tion that equals to d(x;M) for the point inside M and

−d(x;M

k

) otherwise. The medial axis can be consid-

ered as the mean shape of the two active contours and

it is dynamic in the sense that it also evolves as the ac-

tive contours evolve. The directions of the forces im-

posed on the active contours are indicated as arrows

in Figure 5. It can be seen that the forces tend to move

the contours towards their dynamic mean shape.

This approach can be easily extended to the case

of more than two curves as long as the mean shape of

multiple curves can be properly deﬁned.

3 EXPERIMENTAL RESULTS

A couple of experiments on real medical images, us-

ing medial axis as topology constraint and dynamic

shape constraint respectively, are presented in this

section.

In the ﬁrst experiment, the objective was to seg-

ment two ventriclesas well as the entire heart from the

MRI image of a mouse shown in the top-left image of

A MULTIPHASE ACTIVE CONTOUR MODEL WITH DYNAMIC MEDIAL AXIS CONSTRAINT FOR MEDICAL

IMAGE SEGMENTATION

519

Figure 6: Original MRI image of mouse heart (left top); its

boundary term (right top) and regional terms (bottom). (Im-

age was kindly provided by the FUGE Molecular Imaging

Centre, Trondheim, Norway.)

Figure 6. The rest of the Figure shows the three im-

ages, obtained from the MRI image, encoding bound-

ary and regional terms (as g

k

and P

k

in Equ. (5)).

From the image slice, it can be seen that the objects to

be segmented have the same topological structure as

those in Figure 4, i.e., the heart includes both ventri-

cles and the ventricles are mutually exclusive. There-

fore three curves with the same topological structure

were initialized as shown in the ﬁrst image of Fig-

ure 7. For comparison, curve evolution processes with

and without medial axis constraint are shown in Fig-

ure 7 and Figure 8 respectively. Curve evolution with-

out medial axis constraint clearly failed as the outer

contour collapsed, due to lack of well-deﬁned infor-

mation of the outer heart wall, resulting in topological

changes.

iter. # 0 iter. # 5 iter. # 10

iter. # 15 iter. # 20 iter. # 30

Figure 7: Curve evolution without medial axis constraint.

In the second experiment, two knee images were

extracted from a volumetric MRI data. The objec-

tive was to segment bones from these images. The

original images as well as the processed images used

for regional and boundary terms are shown in Fig-

ure 9. Figure 10 shows the curve evolution when the

segmentation is performed independently. It can be

seen that the results for both images are not accept-

able — in the ﬁrst image, leakage happened due to

weak boundary; in the second image, the resulting

iter. # 0 iter. # 5 iter. # 10

iter. # 15 iter. # 20 iter. # 30

Figure 8: Curve evolution with medial axis constraint.

Figure 9: First row: image 1; second row: image 2.

Columns from left to right: original image, regional term,

boundary term.

segmentation failed to include the lower right corner

of the upper bone due to dark shadow. As shown in

Figure 11, much better results were achieved by us-

ing the medial axis constraint with the two evolving

contours helping each other to converge to the correct

bone structures.

4 CONCLUSIONS

This paper proposes a multiphase active contour

method for simultaneous segmentation of multiple

objects in one or multiple images. In the method,

each object to be segmented is associated with an

active contour. The evolution PDE associated with

each active contour involves two main components —

the component responsible for analysis of the image

and each separate contour properties, and the com-

ponent encoding interaction between different evolv-

ing curves. For the former component, a model using

both boundary and regional information from input

images is proposed in order to achieve robust and ac-

curate results. In the model, medial axis plays an es-

sential role in composing the interaction components

to impose topology constraint or dynamic shape con-

straint to achieve robust results.

VISAPP 2010 - International Conference on Computer Vision Theory and Applications

520

iter. # 0

iter. # 0

iter. # 10

iter. # 10

iter. # 20

iter. # 20

iter. # 30

iter. # 30

iter. # 40

iter. # 40

iter. # 50

iter. # 50

Figure 10: Curve evolution without medial axis constraint.

iter. # 0

iter. # 0

iter. # 10

iter. # 10

iter. # 20

iter. # 20

iter. # 30

iter. # 30

iter. # 40

iter. # 40

iter. # 50

iter. # 50

Figure 11: Curve evolution with medial axis constraint.

ACKNOWLEDGEMENTS

The work done in this paper has been supported

from the MEGURATH project (EPSRC project No.

EP/D077540/1).

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A MULTIPHASE ACTIVE CONTOUR MODEL WITH DYNAMIC MEDIAL AXIS CONSTRAINT FOR MEDICAL

IMAGE SEGMENTATION

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