A New Level Set Method for Image Segmentation Integrated with FCM

Zhenping Xie Shitong Wang School of Information, Southern Yangtze University

xiezhenping@yahoo.com.cn

Abstract

It is well known that fuzzy clustering and level set are two fundamental methods for image segmentation. The former focuses on the statistical properties of image features, while the latter aims to acquire the good geometrical continuity of segmentation boundaries. Obviously, two kinds of methods may complement each other. Inspired by this idea, a new level set model integrated with fuzzy c-means (FCM) clustering FCMLS is presented in this paper, where three new strategies are proposed. FCMLS has some remarkable characteristics and better performance in some sense. Furthermore, it can be proved that the new strategies are also available for integrating many popular fuzzy clustering algorithms into various level set models. The results on theoretical and experimental analysis demonstrate the above conclusions on new model. 1. Introduction

Nowadays, the clustering [1,2] and level set methods [3,9,10,11,12,14] are two types of fundamental tools for image segmentation. The former focuses on the statistical properties of image features, while the latter interests in achieving the constraints on boundary smoothness. However, almost all methods existed in the literatures are still unable to stress the two sides simultaneously. Motivated by this idea, a new level set model integrated with fuzzy c-means (FCM) clustering, shortened form FCMLS is proposed in this paper.

An earliest form of level set is Mumford-Shah image segmentation model [3,4], which impose constraints on segmentation boundaries. After this, numerous similar models [10,11,12,13] and their variants [14,15] were presented in the past few years. Similar to many pattern recognition methods, the level set is also an approach of minimizing (or maximizing) an objective function, whose main components includes: region energy term, constraint of the length of the segmentation boundaries, attractive term of image edge and shape constrained term etc. In all of

these terms, the region energy term play a crucial element and determine the primary direction of evolutionary level set. In FCMLS, the modified region energy term integrating with FCM is proposed and three strategies are proposed to implement above task. 2. Level Set Model for Image Segmentation

The earliest level set model was presented in [3,4], which can be expressed as the following problem:

2

\,inf ( , ) ( )u

E u u I dx u dx dsµ νΩ Ω Γ ΓΓ

Γ = − + ∇ +∫ ∫ ∫ (1)

where 0µ > and 0ν > are blending parameters. I represents the original image and u is an optimal piecewise smooth approximation of I ; Γ is the set of segmentation boundaries. Supposed that u is a piecewise constant approximation of I , the above model can be transformed as follows [9,10]:

2

,inf ( , ) ( ( ) )

iiu i

E u I u dx dsνΩ ΓΓ

Γ = − +

∑ ∫ ∫ (2)

For above problem, the level set framework is used to contend with, in which the 1-D contour is embedded into the image domain as the zero-level line of an artificial level set function :φ Ω → . In terms of level set framework, we have the following formulation instead of (2) that are restricted to two regions.

2

,

2

inf ( , , ) ( ( ) ) ( )

( ( ) ) (1 ( )) ( )

uE u u I x u H dx

I x u H dx H

φ φ

φ ν φ

− + −ΩΓ

+Ω Ω

= −

+ − − + ∇

∫

∫ ∫ (3)

where φ represents the regions −Ω for 0φ < and +Ω for 0φ > , as well as the Γ for ( ) 0φ Γ = . The heaviside function ( ) 1H φ = for 0φ < and ( ) 0H φ = for 0φ > is used to distinguish the two regions. In general, the H is replaced by a normalized version Hε to ensure the function to be differentiable, for details see [10].

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3. Level Set Model Integrated with FCM

With human vision, the foreground and background could be extracted in the first step for almost all image segmentations and the refined results could be achieved according to the hierarchical approach. So, the two-regional segmentation is a primary task in image segmentation. Besides, the foreground (or background) should consider to be approximated with several prototypes in real applications. Motivated by this standpoint, the extension of level set model is firstly proposed as follow.

2

,1

2

1

inf ( , ) ( )

( )

C

jx jxjx

C C

kx kxk Cx

E S d dx

S d dx dsν

−

−

− +

−+

Γ=∈Ω

+

Γ= +∈Ω

Γ =

+ +

∑∫

∑∫ ∫

VV

(4)

where C− and C+ are the prototype number of foreground and background with natural number.

1, 2,... , 1,..., lxS l C C C C− − − += + + is 1 when ( )I x can be approximated by prototype l (noted as lv generally)

，otherwise 0. Obviously, there has1

1, C C

lxj

S x− ++

== ∀∑ .

lxd denotes the distance measure. In (4), lxS is unknown variable and can be viewed

as the extract membership of ( )I x to prototype l . Intuitively, it may be extended to fuzzy forms. Substituting the jxS and kxS with m

jxu and mkxu ,

reformed (4) reads

2

, ,1

2

1

1 1

inf ( , , ) ( )

( )

. . 1, ; 0, ,

Cmjx jx

jx

C Cmkx kx

k Cx

C C C

jx kx lxj k C

E u d dx

u d dx ds

s t u u x u l x

ν

−

−

− +

−+

− − +

−

Γ=∈Ω

+

Γ= +∈Ω

+

= = +

Γ =

+ +

+ = ∀ ≥ ∀

∑∫

∑∫ ∫

∑ ∑

V UV U

(5)

where jxu and kxu are fuzzy memberships. To obtain the reasonable jxu and kxu , the constraint of objective of FCM (all objectives described in [1] also can be utilized) may be introduced into the (5), which reads.

2 2

, 1 1

1 1

inf ( , ) ( )

. . 1,

C C Cm m

FC jx jx kx kxj k Cx

C C C

jx kxj k C

E u d u d dx

s t u u x

− − +

−

− − +

−

+

= = +∈Ω

+

= = +

= +

+ = ∀

∑ ∑∫

∑ ∑

V UV U

(6)

In addition, we expect that the prototypes in the same region are as compact as possible while the prototypes in the different regions are as disperse as possible, from which another new constraint is introduced as follow:

2

,1 1

2

1 1

inf ( , ) ( ( ) )

( ( ) )

C C Cm

sep kx jxj k Cx

C C Cmjx kx

k C jx

E u d dx

u d dx

− − +

−−

− + −

−+

+

= = +∈Ω

+

= + =∈Ω

=

+

∑ ∑∫

∑ ∑∫

V UV U

(7)

In terms of the model (4), the above objective should tend to 0 in the ideal case. Summarizing the (5), (6) and (7), the FCMLS model can be described using Lagrange multipliers in total as follows.

2

, , 1 1

2

1 1

2 2

1 1

inf ( , , ) ( ( ) )

( ( ) )

( )

C C Cm mjx r kx jx

j k Cx

C C Cm mkx r jx kx

k C jx

C C Cm m

c jx jx kx kxj k Cx

E u u d dx

u u d dx

u d u d dx ds

λ

λ

λ ν

− − +

−−

− + −

−+

− − +

−

+

Γ = = +∈Ω

+

= + =∈Ω

+

Γ= = +∈Ω

Γ = +

+ +

+ + +

∑ ∑∫

∑ ∑∫

∑ ∑∫ ∫

U VU V

1 1

. . 1, ; 0, ,C C C

jx kx lxj k C

s t u u x u l x− − +

−

+

= = +

+ = ∀ ≥ ∀∑ ∑

(8)

where rλ and cλ are Lagrangian coefficients, in general 1rλ = . Similar to above fashion, the following necessary conditions can be deduced (the details are omitted for the sake of the limitation of paper length).

11

2 2

1

1 2 2

1

11

2 2

1

1 2 2

1

( ( )) (1 ( ))

( ( )) (1 ( ))

( ( )) (1 ( ))

( 1 ( )) ( )

C C m

c jx lxCl CC C

pc px lx

l C

jxC C m

c jx lxCl CC

q Cc qx lx

l

H d H d

H d H d

u

H d H d

H d H d

λ φ φ

λ φ φ

λ φ φ

λ φ φ

− +

−−

− +

−

− +

−−

−−

+ −

= ++

=

= +

+ −

+= +

= +

=

+ + −

+ + − =

+ + −

+ + − +

∑∑

∑

∑

∑

1

11

2 2

1

2 2

1

2 2

1

2 2

1

, ,

( 1 ( )) ( )

( ( )) (1 ( ))

( 1 ( )) ( )

( 1 ( )) ( )

C

C m

c kx lxlC C

c px lxl C

kx

C

c kx lxlC

c qx lxl

j x

H d H d

H d H du

H d H d

H d H d

λ φ φ

λ φ φ

λ φ φ

λ φ φ

+

−

− +

−

−

−

−

−

=+

= +

=

=

∀

+ − +

+ + − =

+ − +

+

+ − +

∑

∑

∑

∑

∑

1

111

1

, ,C

pm

C C

q C

k x−

− +

−

−

=−

+

= +

∀

∑

∑ (9)

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1

1

1

1

(( ( )) ( ) ) ( ),

(( ( )) ( ) )

(( 1 ( )) (1 ( )) ) ( ),

(( 1 ( )) (1 ( )) )

C Cm m

c jx lxl Cx

j C Cm m

c jx lxl Cx

Cm m

c kx lxlx

k Cm m

c kx lxlx

H u H u I x dxj

H u H u dx

H u H u I x dxk

H u H u dx

λ φ φ

λ φ φ

λ φ φ

λ φ φ

− +

−

− +

−

−

−

+

= +∈Ω+

= +∈Ω

=∈Ω

=∈Ω

+ += ∀

+ +

+ − + −= ∀

+ − + −

∑∫

∑∫

∑∫

∑∫

v

v

(10)

where (9) is valid only if 1m > . Moreover, the evolutional equation corresponding to φ reads

2

1 1

2

1 1

( 1) ( ) ( )[ (( 1) )

(( 1) ) ( )]

C C Cm m

c jx kx jxj k C

C C Cm m

c kx jx kxk C j

t t t u u d

u u d

εφ φ δ φ λ

φλ νφ

− − +

−

− + −

−

+

= = +

+

= + =

+ = + ∆ ⋅ + +

∇− + + + ⋅∇∇

∑ ∑

∑ ∑(11)

where t∆ is the time step. In summary, the FCMLS algorithm can be proposed as follows. FCMLS Algorithm

1

Set the values of m , cλ , C− and C+ , initialize the segmentation boundary and level set φ , initialize

, ,j k j k∀v v in terms of segmentation result, and set 0t = ;

2 Update , , ,jx kxu u j k x∀ using (9);

3 Calculate ( 1)tφ + using (11); 4 Update , ,j k j k∀v v using (10);

5 If the stop conditions are satisfied, then terminate; otherwise set 1t t= + and go to Step 2.

Remarks: The above FCMLS algorithm is a standard procedure. Furthermore the fast version can be implemented using the “band level set” [5] and “fast marching” [6]. 4. Model Analysis of FCMLS 4.1. Characteristics of FCMLS

Characteristic 1 ： FCMLS has more perfect mathematic form than existed level set methods.

For existed level set models, some key variables such as iu in (2) cannot be solved directly from the models themselves. Usually, by introducing the empirical definitions a crude method is adopted therein. In contrast, the solutions of key variables in FCMLS can be directly derivate from the necessary condition of minimizing objective function, rather than rely on other empirical knowledge.

Characteristic 2 ： FCMLS can pay equivalent attention to global and local features of an image simultaneously.

For level set model, its main contribution to image segmentation is what can gain the good geometrical continuity of segmentation boundaries. However, the global statistical features of image pixel cannot be contended with well. Inversely, the image segmentation methods based on the (fuzzy) clustering is good at the analysis of global statistical features of image pixel, while it cannot ensure the geometrical continuity of segmentation boundaries. Inspired by the above understanding, two approaches of combining the above two methods have been proposed in [7,8], but they have not put forward a unified model. Obviously, the FCMLS model is a general model combined with above two methods. That is to say, the global and local features of an image can be considered in equity.

Characteristic 3 ： Likely to level set model, FCMLS still has the great expansibility.

Nowadays, many variants of original level set methods are implemented by introducing some enhanced strategies such as boundary smoothness [14], image texture feature [13], the prior constraint of object shape [15] and the manifold metric [16] etc. On the other hand, because the FCMLS is implemented only by combined with a basic level set method and FCM, above all enhanced strategies also can be introduced into our novel model. 4.2. Remarks on Model Parameters of FCMLS

For parameters C− and C+ , there has no general optimal value due to the huge diversities of different images. So it is difficult to discuss the parameter selection, which might be examined in the practical applications and some discussion will be performed in section 5.

Parameter m in FCMLS plays the same role as fuzzy exponent of fuzzy clustering, so some analysis of fuzzy exponent in the literatures might be referenced. Based on the previous researches, 2m = may be satisfied with almost all applications. In our paper, the same setting is adopted. Parameter cλ can be viewed as the weighted factor between the objectives of FCM and level set. Obviously, the larger value will results in the stronger constraint of fuzzy clustering on total model. While, it must be noticed that even if the very large value of cλ is adopted, the segmentation results obtained by FCMLS still have some difference from those results obtained by FCM. It accords with our expectation and is illustrated by the following experimental results.

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From our experimental comparisons, the default parameter is fixed to 50cλ = . 5. Experiments Studies

To further investigate the performance of FCMLS, several image segmentation results are reported in this section. In the following experiments, the HSV and Lab color space features are chosen for color images in 5.1 and 5.2 respectively. 5.1. Special Image Segmentation

In this subsection, several special image segmentation experiments are studied to evaluate the performance of FCMLS. Here special image segmentation mainly means that the image requires to be segmented into two parts, foreground and background, and the C− , C+ are known or can be predicted in reason. This task is a primary segmentation process in image segmentation. It is just the segmentation task in many cases, and in other cases the segmentation task also can be achieved according to recursive such process. Here several peach shadow image segmentations are examined.

(a) (b) (c) (d) (e) Figure 1. The peach shadow image segmentation experiment.

Figure 1 illustrates the several peach shade image segmentation results obtained by four algorithms including FCMLS ( 1 2C C− += = ), FCM (the number of cluster is set to 3), ITCE [12] and model (3) (shortened by BMS). Where the initial level sets of FCMLS, ITCE and BMS are listed in the first column. Here ITCE is a typical level set method that the region energy term is the metric function of probability of the pixel’s features, which represents the latest efforts in level set methods for image segmentation.

From the Figure 1, we can find that the FCMLS get the satisfied segmentation results in all peach shade

images, while others cannot do it. Moreover, from the detailed comparisons between the columns (b) and (c) in Fig. 1, it is clear that the segmentation results obtained by FCMLS are similar to the segmentation results obtained by FCM in the mass, but the better detailed results are gained due to combining with the advantages of level set model. For FCM, because its clustering result is sensitive to the noise points of the original images, it cannot gain the accurate segmentation boundary between the peach part and the shade part.

Based on the above experimental results, two obvious advantages of FCMLS are exhibited. Firstly, the FCMLS exactly gain the combined merits of fuzzy clustering and level set model. Secondly, for special image segmentation problem FCMLS has higher effectiveness and robustness than previous algorithms. 5.2. Natural Image Segmentation

Figure 2. Several natural image segmentation results obtained by FCMLS with different prototype numbers

When FCMLS are applied to natural image segmentation, the different ,C C− + should be tested because of its complexity and diversity. Here, some segmentation comparisons are reported on three standard natural images displayed in Figure 2. Where, the first image is selected from [14], and the others are selected from Berkeley standard image segmentation database [17]. The corresponding segmentation results

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obtained by FCMLS with different prototype number are illustrated in Fig. 2, where the values of C− and C+ are given therein.

For first image, the similar segmentations are obtained by FCMLS with different parameter settings of ,C C− + . And the different segmentation results are obtained for other two images. However, all the segmentation results in Fig.2 obtained by FCMLS can be looked as the reasonable segmentations, because that several segmentation results also could be gained by human vision for most natural images. Furthermore, it can be found that the segmentation results depend on the relative value of C− and C+ , but not the number value. Evidently, it’s an exciting discovery, which will help us easily choose the suitable value of C− and C+ in some sense. 6. Conclusions

In this paper, a new level set model integrated with FCM is presented. The results on theoretical and experimental analysis demonstrate the rationality and effectiveness of the proposed model, whose several merits can be concluded as follows:

1. FCMLS has more perfect mathematic form than existed level set models.

2. FCMLS can pay equivalent attention to global and local features of an image simultaneously.

3. FCMLS can get better performance in some special image segmentation problems.

4. Likely to level set model, FCMLS still has the great expansibility.

Although some advantages of FCMLS have been gained, how to determine the optimal parameter values of C− and C+ deserve to be further studied. It is appealing that some strategies to determining the optimal cluster number in existed clustering methods can be utilized. In the near future, we will study this problem in depth.

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