Computer Vision and Image

Understanding

Manuscript Draft

Manuscript Number: CVIU-12-298

Title: Lattice Boltzmann Model for Fast Level Set Algorithm Using the

Multiple Kernel Fuzzy C-Means

Article Type: Regular Paper

Keywords: Lattice Boltzmann method, multiple kernel fuzzy c-means, level

set equation, image segmentation, texture image

Abstract: Over the last decades, the development of high dimensional

large-scale imaging devices increases the demand of fast and accurate

image processing techniques. Due to its intrinsic advantages which allows

to handle complex shapes and topological changes, the level set method

(LSM) is an promising technique but is computational expensive. As a fast

alternative approach for solving the level set equation, the highly

parallelizable lattice Boltzmann method (LBM) has attracted much

attention. Nevertheless, nearly all the level set image segmentation

methods based on LBM employ the Bhatnagar-Gross-Krook (BGK) collision

model. In this paper we firstly demonstrate that there is no necessary to

use the BGK model in the level set image segmentation and the method is

faster with almost the same result when considering zero collision.

Experimental comparison results with four image segmentation methods

based on level set using LBM-BGK confirm this theoretical analysis. We

secondly propose a fast and efficient LBM-based level set method which

incorporates intensity and texture information of the image. From the

partition matrix of a multiple kernel fuzzy c-means (MKFCM), we design a

multiple kernel fuzzy stop function (MKFSF) for the LBM solver based on

the zero collision model. The method is fast, accurate and highly

parallelizable. Experiments on natural and medical images demonstrate the

superiority of the proposed method in term of speed and efficiency

comparing with five image segmentation methods based on level set.

Highlights:

We demonstrate that there is no necessary to use the BGK model in the level set image segmentation.

We propose a fast and efficient LBM-based level set method which incorporates intensity and texture

information of the image.

We design a multiple kernel fuzzy stop function for the LBM solver based on the zero collision model.

*Highlights (for review)

1  

Lattice Boltzmann Model for Fast Level Set Algorithm Using the Multiple Kernel Fuzzy C-Means

 Souleymane Balla-Arabé, Xinbo Gao*

School of Electronic Engineering, Xidian University, Xi’an 710071, China

Abstract Over the last decades, the development of high dimensional large-scale imaging devices increases the demand

of fast and accurate image processing techniques. Due to its intrinsic advantages which allows to handle complex

shapes and topological changes, the level set method (LSM) is an promising technique but is computational

expensive. As a fast alternative approach for solving the level set equation, the highly parallelizable lattice

Boltzmann method (LBM) has attracted much attention. Nevertheless, nearly all the level set image segmentation

methods based on LBM employ the Bhatnagar-Gross-Krook (BGK) collision model. In this paper we firstly

demonstrate that there is no necessary to use the BGK model in the level set image segmentation and the method

is faster with almost the same result when considering zero collision. Experimental comparison results with four

image segmentation methods based on level set using LBM-BGK confirm this theoretical analysis. We secondly

propose a fast and efficient LBM-based level set method which incorporates intensity and texture information of

the image. From the partition matrix of a multiple kernel fuzzy c-means (MKFCM), we design a multiple kernel

fuzzy stop function (MKFSF) for the LBM solver based on the zero collision model. The method is fast, accurate

and highly parallelizable. Experiments on natural and medical images demonstrate the superiority of the proposed

method in term of speed and efficiency comparing with five image segmentation methods based on level set.

2012 Elsevier Science B.V. All rights reserved

Keywords: Lattice Boltzmann method, multiple kernel fuzzy c-means, level set equation, image segmentation,

texture image 1. Introduction

Image segmentation is a sine qua non step for computer vision systems. It aims to partition a given image into several distinct regions or to detect an object in a given scene. This task is more challenging that there is no a general method which is effective for all kind of images, so one should choose the proper method according to the characteristics of the image in hand.

The level set method (LSM) ([49]-[50]) is part of the sub-family of active contours models (ACMs) ([17], [36]-[39]) which uses the Eulerian framework, i.e., the geometric representation of the active contour instead of the parametric representation, i.e., the Lagrangian framework ([35], [13], [21]). In two-dimensional (2D) space, the basic idea of LSM is to evolve a given curve toward its interior or exterior normal until defining the boundary of the object of interest. The curve evolution is driven by the level set equation (LSE) which is a partial differential equation and, in its general form can be expressed as

⎛ ⎞∂ ∇= ∇ + ∇ ⋅⎜ ⎟⎜ ⎟∂ ∇⎝ ⎠

Vtφ φφ α β

φ, (1)

                                                              * Corresponding author: Tel.: +862988201838; fax: +862988201620. E-mail address: xbgao@ieee.org

*2. Manuscript

2  

where φ is the level set function (LSF), V is the speed function which drives the active contour

towards the region boundaries, the second term in the brackets of the right hand stands for the curvature and is used to smooth the contour, α and β are user-controlling parameters. Usually two

approaches are employed to stop the evolving curve on the boundary of the desired object. The first one uses an edge indicator depending on the gradient of the image like in classical snakes and active contours models ([22]-[26], [30]); and the second one uses some regional statistics to stop the evolving curve on the actual boundary ([27], [28], [31]). The latter is more robust against noise and can detect objects with weak boundaries and without edges. One of the most used region-based technique was proposed in [29] where Chan and Vese introduced a level set formulation to minimize the Mumford and Shah functional [32] that converted the problem into a mean curvature flow problem like in the active contours but the results is more effective than the classical active contours because the stopping term did not depend on the gradient of the image which reduces the dependency on clear edges. However, firstly the method is sensitive to the position of the initial contour, and the evolving curve can easily be trapped into local minima. Secondly, the method is not suitable for parallel programming because at each iteration the average intensities inside and outside the contour should be computed, which increases enormously the CPU time by increasing communications between processors. And finally, the method is not effective for texture image segmentation. To this end, we propose a new method which tries to handle the above disadvantages. Our method is based on the idea which aims to stop the evolving curve according to the membership degree of the current pixel to be inside or outside to the active contour. The membership degree is computed according to the intensity of the pixel and the texture information using the multiple kernel fuzzy c-means (MKFCM) algorithm [11].

In the level set method, the movement of the zero level set is driven by the LSE, which is a partial differential equation (PDE). For solving the LSE most classical methods use some expensive finite difference, finite element or finite volume approach and an explicit computation of the curvature [33]. Unfortunately, these methods cost much more CPU time. Recently, the lattice Boltzmann method (LBM) is used as an alternative approach for solving LSE ([1], [14]-[16], [48]). It can better handle the problem of time consuming because the curvature is implicitly computed, the algorithm is simple and highly parallelizable. However, nearly all these level set image segmentation approaches based on LBM use the Bhatnagar-Gross-Krook (BGK) collision model. In this paper we firstly demonstrate that there is no necessary to use the BGK model for level set image segmentation, and the method can be faster with the same quality of segmentation result when using the zero collision model.

Secondly, we propose a level set method based on LBM for texture image segmentation. In our proposed method, using the multiple kernel fuzzy c-means (MKFCM) algorithm, we design a multiple kernel fuzzy stop function (MKFSF). The method is fast, efficient when detecting objects with weak or without edges, and effective for texture images. It has firstly, the advantage of the MKFCM which allows it to combine simultaneously the intensity and the texture information in order to decide to stop or not the evolving curve according to the membership degree of the current pixel; secondly the advantages of the level set method which allows it to handle complex shapes, topological changes, different constraints on the contours smoothness, speed, size and shape are easily specified; thirdly the advantages of the lattice Boltzmann method which makes it very suitable for parallel programming due to its local and explicit nature.

The rest of this paper is organized as follows: In section II, a general overview of the LBM and the MKFCM techniques is presented. Section III demonstrates that the BGK collision model is not necessary for level set image segmentation. Section IV explains the formulation of the proposed fast

3  

image segmentation method based on the modified level set. Section V validates the theoretical analysis and the performance of the proposed method through experimental results. And section VI concludes the paper. 2. Background

This section gives a general idea of the lattice Boltzmann model and the multiple kernel fuzzy c-means algorithm.

2.1 Lattice Boltzmann model

The LBM is firstly designed to simulate Navier-Stokes equations for an incompressible fluid [2]-[4]. The evolution equation of LBM is

( , 1) ( , ) ∂⎛ ⎞+ + − = ⎜ ⎟∂⎝ ⎠r r r

i i icoll

ff r e t f r tt

, (2)

where if is the particle distribution function and ( )∂ ∂ collf t is, in this paper, the

Bhatnager-Gross-Krook (BGK) collision model ([5], [7]-[9]) with a body force rF .

21 [ ( , ) ( , )] .∂⎛ ⎞ = − + ⋅⎜ ⎟∂⎝ ⎠

rr r reqi i i

coll

f Df r t f r t F et bcτ

, (3)

where D is the grid dimension, b is the link at each grid point, c is the length of each link which is set to 1 in this paper, τ represents the relaxation time and eqif the local Maxwell-Boltzmann equilibrium particle distribution function expressed in its continuous form as

3 2 2(2 ) exp ( ) 2− ⎡ ⎤= − −⎣ ⎦r reqf RT u RTρ π υ , (4)

where rυ is the particle velocity and ru the macroscopic velocity. The equilibrium distribution can be

expressed in discrete form as follows when modeling typical diffusion phenomenon,

( ) = =∑eqi i ii

f A with fρ ρ ρ , (5)

where ρ is the macroscopic fluid density. By performing the Chapman-Enskog expansion [10], the

following diffusion equation can be recovered from LBM [3],

( )∂ = ∇ +∂

div Ftρ β ρ . (6)

Substituting ρ by the signed distance function φ in Eq. (5), the LSE can be recovered. In our model we use the D2Q5 ( 2=D , 5=b ) LBM lattice structure. The body force F acts as the image data link for the LBM solver.

Reference [1] used another approach to perform the level set image segmentation. The Eq. (3) is the general evolution equation of LBM. However, in level set based image segmentation a stop function

( )rg r is necessary for stopping the evolving curve or surface at the boundaries of the object. In order

to introduce the stop function into LBM, they considered a medium between the nodes of the lattice. The particles can pass through the medium with a possibility of ( )rig r , and will be punched back where they were with a possibility of 1 ( )− rig r . The LBM evolution equation is modified as

4  

1, 1 )[ , [( , , ] ]

(1

) ( ( ) ( ) ( )

() )( ) , ,

+ + = + − +

+ − +

r

r

r r r r r

r r

eqi i i i i i

i i i

f r e t g r f r t f r t f r t

g r f r e t

στ (7)

where σ is the convection coefficient and the macroscopic fluid density ρ is set as a signed

distance function. More details about this approach can be found in [1].

2.2 Multiple kernel fuzzy c-means algorithm

As an enhancement of the fuzzy c-means algorithm ([6], [11], [40], [41]), The multiple kernel fuzzy c-means aims to minimize the following objective function

2

1 1

1

( )

. . 1 ,

= =

=

= −

=

∑∑

∑

c nmij com j i

i j

c

ijk

Q u x o

s t u

ϕ

(8)

where io is the cluster center, m the fuzzification parameter, comϕ a transformation function defined by a combination of kernels

( , ) ( ) , ( )com com comk x y x yϕ ϕ= , (9)

which verify the following relations. If 1k and 2k are kernels over ,pRΩ×Ω Ω⊆ then

(1) 1 2( , ) ( , ) ( , )= +comk x y k x y k x y is a kernel,

(2) 1( , ) * ( , )=comk x y k x yα is a kernel, when 0α > ,

(3) 1 2( , ) ( , )* ( , )=comk x y k x y k x y is a kernel.

The multiple kernels technique offers many advantages, such as the ability to combine many kind of information. In our method we combine a kernel for intensity information and a kernel for texture information in order to obtain an efficient image segmentation method. 3. The Lattice Boltzmann Method Without BGK Collision Model

Nearly all the level set image segmentation methods based on LBM employ the Bhatnagar-Gross-Krook (BGK) collision model. In this section we demonstrate that there is no necessary to use the BGK collision model when solving the LSE for image segmentation using the lattice Boltzmann method.

Let the level set function φ be a signed distance function. In level set image segmentation method,

by considering the motion by mean curvature, the velocity field which guides the evolution of the

5  

active contour, contains only a component in the normal direction [33], i.e., the tangential component is identically zero. Let set to the limit conditions of this velocity field equal to zero; when updating φ

on the entire image domain, all the contours (not only the zero one) evolve in the normal direction. By analogy, when using the lattice Boltzmann method to update the level set function the entire particles move in the normal direction and are subjected to the same velocity field, i.e., there are almost no collisions between them. Thus we can set the collision operator to zero in the LBM evolution equation. Furthermore, in image segmentation we are just interested by the final result, i.e., the thermodynamic equilibrium in the LBM; and the BGK collision operator vanishes at this state.

The above statement can be formulated as follows. Let consider the discrete velocity Boltzmann equation without an external force, which describes the spatiotemporal evolution of the distribution function of particles having a given velocity rυ at a given position and a given time t

.∂ ∂⎛ ⎞

+ ∇ = ⎜ ⎟∂ ∂⎝ ⎠

ri ii i

coll

f ff

t tυ , (10)

the right term of the above equation is the collision operator and represents the effect of collisions between particles. If this term is null, the particles are simply advected and we recover the level set equation by replacing if by the level set function φ

. 0∂ + ∇ =∂

rit

φ υ φ . (11)

Thus, the LSE can be solved using the LBM with zero collision model. By considering the mean curvature motion Eq. (11) can be rewritten as

∂= ∇

∂ itφ υ φ , (12)

which is equivalent to the level set equation Eq. (1) with

∇= + ∇ ⋅

∇iV φυ α β

φ. (13)

The external forces can still be added after the collision step by modifying the distribution function as [47]

2 .← + ⋅r r

i i iDf f F e

bc, (14)

where D is the grid dimension, b is the link at each grid point, c is the length of each link. The macroscopic fluid density ρ can still be recovered using Eq. (5).

The LBM evolution equation ([1], [3], [14]) is therefore simplified. In [14], we can ignore the collision step and directly go to the streaming step after including the external force. In [1], the evolution Eq. (7) is simplified as

6  

( ) ( ) ( )( )

, 1 ( )[ , , ]

, .

+ + = − + +

+ +

r r r r r

rr

ri i i i i i

i i

f r e t g r f r t f r e t

f r e t

σ (15)

The main advantage of this approach is the gain of CPU time; the method becomes faster with almost the same result when performing the image segmentation. Experimental results on four methods ([1], [14]-[16]) confirm this theoretical result. 4. The Proposed Fast Image Segmentation Method Based on Level Set

This section details firstly, the conception and the analysis of the proposed fast image segmentation method based on level set, and secondly the software implementation.

4.1 Design and analysis of the proposed method

Let consider the objective function of the multiple kernel fuzzy C-means (MKFCM)

2

1 1

1

( )

. . 1 .

= =

=

= −

=

∑∑

∑

c nmij com j i

i j

c

ijk

Q u x o

s t u

ϕ

(16)

To deal with texture information, let set jx in (16) as 3[ , ' , ]j j j jx y y s R−= ∈ , in which jy R∈ is

the intensity of pixel j . The 2-tuple 2[ ' , ]j jy s R− ∈ is a simple descriptor is a simple descriptor of

the texture information at pixel j , where ' jy − is the filtered intensity of pixel j and js is the

standard variance of the intensities of the pixels in the neighborhood of pixel j ([11], [19], [42]).

Then we define the combined kernel as

1 2*comk k kα= + , (17)

where 1k is the Gaussian kernel for pixel intensities and 2k the Gaussian kernel for texture information

21( , ) exp( )i j i jk x x r y y= − − , (18)

2

2( , ) exp( [ ' , ] [ ' , ] )i j i i j jk x x r y s y s− −= − − . (19)

By considering the constrained case which assumes that the clusters centers in the kernel space as the points that are mapped from points in the original data space, the standard Euclidean distance

2( ) ( )com j com ix oϕ ϕ− becomes

7  

2( ) ( ) ( ). ( ) ( ). ( )

2 ( ) ( )

2(1 ( , )),

− = +

−

= + −

com j com i com j com j com i com i

com j com i

com j i

x o x x o o

x o

k x o

ϕ ϕ ϕ ϕ ϕ ϕ

ϕ ϕ

α

(20)

and the objective function is then replaced as

1 1

1

(1 ( , ))

. . 1 .

= =

=

= + −

=

∑∑

∑

c nmij com j i

i j

c

ijk

Q u k x o

s t u

α

(21)

By analogy with [20], 1 ( , )com j ik x oα+ − can be seen as a robust new distance measurement derived

from the kernel space. For solving the minimization problem of Q , we should iteratively update the

cluster centers and memberships according to some necessary conditions that we obtain by computing

the first derivative of Q with respect to iju and the first derivative of Q with respect to io ; and

then setting these derivatives equal to zero. These necessary conditions are

1( )1

1( )11

(1 ( , ))

(1 ( , ))

mcom j i

ij c mcom j ll

k x ou

k x o

α

α

−−

−−

=

+ −=

+ −∑, (22)

1

1

( , )

( , )

n mil com l i ll

i n mil com l il

u k x o xo

u k x o=

=

=∑∑

. (23)

Now let consider the approach used in [1] to solve the level set equation. As we demonstrate in

Section III, the evolution Eq. (7) is simplified as

, 1 ( )[ , ,( ) ]( ) (.

)( , )

+ + = − + +

+ +

r rr r

r

r r

ri i i i i i

i i

f r e t g r f r t f r e tf r e t

σ (24)

Here, for a two phases level set case, we design a multiple kernel fuzzy stop function (MKFSF) which will stop the evolving curve on the boundaries of the object as

1 2

1 2 1 , 0 1 , 0.

= −

+ = ∀ ≤ ≤ ∀ >

nj j

j j kj

mkfsf u u

with u u x y u k j and n (25)

Then the new LBM evolution equation is:

8  

1 2( , 1) *[ ( , ) ( , ) ]

( , ).

+ + = − − + +

+ +

r r r r r

r r

ni i j j i i i

i i

f r e t u u f r t f r e t

f r e t

σ 26)

When analyzing Eq. (29) we can see that

0 1mkfsf≤ ≤ . (27)

The active contour will evolve when 1 ju and 2 ju are different, and will stop when

1 2j ju u≈ , (28)

because MKFSF will be near to zero. The parameter n can be used to increase the convergence toward zero of the MKFSF.

4.2 Implementation

The main steps for the algorithm implementation are outlined as follows:

1- Initialize if as a signed distance function and class centroid values 1o and 2o ;

2- Compute 1 ju and 2 ju with Eq. (22);

3- Calculate the MKFSF function with Eq. (25) ;

4- Perform the modified LBM evolution equation Eq. (26);

5- Find the contour the zero contour of any if for 0≠i , which correspond to the zero level of

the level set function φ ;

6- If the segmentation is not done update 1o and 2o with Eq. (23) and go to step 2. 5. Experiments and Analysis

This section is divided into three parts. The first part confirms the theoretical prevision of section III on four level set image segmentation methods based on LBM-BGK. The second part demonstrates the proposed method in terms of speed and effectiveness, and also confirms the theoretical analysis of section III in term of efficiency. In the last part, we use the Hausdorff method to evaluate objectively the performance of the proposed image segmentation method.

All the methods have been implemented using Matlab R2010b installed on a PC AMD Athlon (tm) 5200 processor with a clock speed of 2.31 GHz and 2 GB of RAM. The kernel parameter r is set at

2(150)r −= as suggested in [34].

5.1 Validation of the theoretical analysis of Section III

In this part we demonstrate experimentally the theoretical result obtained in Section III. Using four different level set image segmentation methods based on LBM-BGK, we show that all these methods can be speedup just by considering zero collision instead of using the BGK collision model.

9  

The first method was used by Aaron Hagan and Ye Zhao in [14] for 3D image segmentation. Fig.2 shows the experimental results obtain on the one hand by using the BGK collision model, on the other hand by using the zero collision model. The CPU times are displayed in TABLE I. We can see that all the segmentation results are almost the same with a gain on CPU time when considering zero collision. And this gain on time will be considerably huge when processing volume images since it will increase proportionally to the number of images to be processed.

The second method was used by Yu Chen et al. in [12] where, they considered a medium between the nodes of the lattice in order to introduce the stop function of the evolving curve into the LBM. Fig.3 shows the experimental results and the CPU times are displayed in TABLE II. It can also be seen that by considering zero collision we gain in CPU time with the almost same segmentation result.

The third method is a region based level set image segmentation [16]. The method aims to simulate the Chan and Vese [29] method using the LBM-BGK approach. Fig.4 shows the experimental results. The CPU times can be seen in TABLE III. We can also notice that the zero collision model allows a gain in CPU time with the same final result.

The fourth method is the one we proposed in [15], which aims to design a region based stop function UPF (Unsigned Pressure Function) for the Chen’s model [12]. The experimental results can be seen in Fig.5 and all the CPU times are displayed in TABLE IV. This experiment confirms also theoretical prevision; with zero collision we gain in CPU time with the same final result.

5.2 Comparison in terms of speed and efficiency

In this part we compare the proposed fast level set method with five level set methods for image segmentation ([29], [14], [16], [1], [43]) in order to demonstrate its ability in terms of speed and efficiency.

Fig.6 displays the experimental results on natural and medical images. The first row shows the initial contours. The second row shows the segmentation result of the Chen’s method [1]. The third row shows the segmentation result of the method introduce by Z. Wang et al. in [16]. The fourth row shows the segmentation result of the Li’s method [43]. The fifth row shows the segmentation result of the Chan and Vese method [29]. The sixth row shows the segmentation result of the method used by Aaron Hagan and Ye Zhao in [14]. And the last row shows the segmentation result using the proposed method.

The first and the last column are the segmentation results on medical images with weak edges and where intensity inhomogeneity can be clearly noticed. We can see that the proposed method gives the best results, i.e., the contours are thin and present no discontinuities. Chen’s method for example, based on an edge stopped function is not efficient because of the presence of weak edges. Aaron and Zhao’s method gives a resulted contour with many discontinuities. Furthermore the proposed method detects more useful objects than the Chan and Vese and the Z. Wang et al. proposed methods.

The third column demonstrates the ability, in term of effectiveness, of the proposed method on natural image in presence of intensity inhomogeneity.

The second and the fourth column demonstrate the efficiency of the proposed method to deal with texture information introduce in the multiple kernel fuzzy c-means. It can be seen that its segmentation results are the best one all, and almost all the other methods have failed to detect the rabbit in the fourth column. TABLE V displays all the CPU times, and we can see that the proposed method is the fastest one. The good quality of the proposed method results confirms also the theoretical prevision of Section III in term of effectiveness, since it also considers directly the thermodynamic equilibrium where there are no collisions.

10  

Fig.2. Segmentation result of natural and medical images using the Hagan and Zhao method [14]. The first column shows the initial

contour. The second column the result using BGK. And the third one shows the result when considering zero collision.

TABLE I. CPU TIMES OF THE AARON-ZHAO’S METHOD WITH AND WITHOUT USING BGK.

TABLE II. CPU TIMES OF THE CHEN’S METHOD WITH AND WITHOUT USING BGK.

   

   

   

   

   

CPU time (s) Column I Column II Column III Column IV Column V

With BGK 14.9304 21.9917 1.5914 4.1368 2.4765

without collision 12.7330 19.7174 1.0444 3.0038 1.9427

Image dimension 506 x 427 632 x 378 432 x 355 416 x 355 573 x 302

CPU time (s) Column I Column II Column III Column IV Column V

With BGK 4.1816 3.1722 9.1988 3.0067 3.1471

Without collision 3.0602 2.2693 7.3894 2.0045 2.3276

Image dimension 579 x 435 432 x 355 632 x 378 430 x 349 646 x 407

11  

Fig.3. Segmentation result of natural and medical images using Yu Chen method [12]. The first column shows the initial contour.

The second column the result using BGK. And the third one shows the result when considering zero collision.

5.3 Supervised evaluation of the proposed method

This part evaluates objectively the proposed method using the Hausdorff distance ([44], [45]). The Hausdorff distance measures the similarity between two images. The lower it is, the better is the segmentation result. It is computed as follows

( , ) max( ( , ), ( , )),C ref C ref ref CHAU I I h I I h I I= (29)

TABLE III. CPU TIMES OF THE METHOD PROPOSED IN [16] WITH AND WITHOUT USING BGK.

   

   

 

   

   

CPU time (s) Column I Column II Column III Column IV Column V

With BGK 14.7721 3.6064 21.2392 11.2102 2.1105

Without collision 12.3586 2.0974 18.0199 10.0445 1.6545

Image dimension 579 x 435 558 x 363 632 x 378 432 x 355 557 x 364

12  

Fig.4. Segmentation result of natural and medical images using the method used in [16]. The first column shows the initial contour.

The second column the result using BGK. And the third one shows the result when considering zero collision. where

( , ) max( min )refC

C ref b Ia Ih I I a b

∈∈= − . (30)

Fig. 7 and Fig. 8 show the experimental results when evaluating the proposed method, the

Chan-Vese’s method and the Z. Wang et al.’s method. All the images and the humans’ segmentations used as ground truth are from the Berkeley segmentation dataset BSDS300 [46]. The curves in Fig. 7 show that the proposed method has the lowest Hausdorff distances, i.e., its segmentation results are more similar to the ground truth than the Chan-Vese method and the Z. Wang method. Thus the proposed method is not only fast but also accurate.

   

     

     

   

   

13  

Fig.5. Segmentation result of natural and medical images using the method we proposed in [15]. The first column shows the initial contour. The second column the result using BGK. And the third one shows the result when considering zero collision.

TABLE IV. CPU TIMES OF THE METHOD WE PROPOSED IN [15] WITH AND WITHOUT USING BGK.

6. Conclusion

In this paper, we have firstly demonstrated that by considering zero collision in the lattice Boltzmann approach for level set image segmentation instead of the BGK collision model, we obtain almost the same final segmentation result with a gain in CPU time. This gain will be enormous when processing volume images. We secondly proposed an image segmentation method which incorporate

   

   

   

   

   

CPU time (s) Column I Column II Column III Column IV Column V

With BGK 4.2252 2.2338 2.6925 2.0832 2.2512

Without collision 3.0806 1.2284 2.0059 1.4797 1.5741

Image dimension 632 x 378 646 x 407 646 x 407 646 x 407 646 x 407

14  

Fig.6. Segmentation result of natural and medical images. The first row shows the initial contours. The second row shows the segmentation result of the Chen’s method [1]. The third row shows the segmentation result of the method introduce by Z. Wang et al. in [16]. The fourth row shows the segmentation result of the Li’s method [43]. The fifth row shows the segmentation result of the Chan and Vese method [29]. The sixth row shows the segmentation result of the method used by Aaron Hagan and Ye Zhao in [14]. And the last row shows the segmentation result of the proposed method.

TABLE V. CPU TIMES OF THE PROPOSED METHOD, THE CV’S METHOD, THE LI’S METHOD THE Z.WANG’S METHOD, THE

CHEN METHOD AND THE AARON-ZHAO’S METHOD

CPU times (s) Proposed. Method

CV’s Method

Li’s Method. ( 5t sΔ = )

Z. Wang’s Method

Chen’s Method

Aaron- Zhao’s Method

Image dimensions

Row II 0.2367 71.8206 135.6697 7.5509 1.6151 1.1396 390x311 Row III 0.4055 172.6487 227.0677 18.7619 0.7570 1.0797 436x436 Row IV 1.2291 289.2372 246.3588 39.5608 26.4894 2.5979 774x470 Row V 1.4543 390.5590 341.0133 145.9955 25.1511 3.0260 592x406 Row VI 0.6723 133.5465 161.8987 70.9657 8.7314 1.4763 416x355 Row VII 0.1962 44.7518 97.7718 23.4240 1.3036 0.4902 365x286

15  

Fig.7. Hausdorff evaluation of the proposed method, the Z. Wang et al. method and the Chan- Vese method.

the advantages of the level set method, the lattice Boltzmann method and the multiple kernel fuzzy c-means. The method is fast, firstly due to the low computational complexity of the LBM; and secondly because of the use of the zero collision model which reduces the computational complexity of the LBM. Furthermore due to the local nature of the LBM, the method is highly parallelizable and suitable for parallel programming using graphics processing units for example. The good accuracy of the method comes from the use of the MKFCM which allows it to combine several information such as intensity and texture information. Experimental results on natural and medical images demonstrate subjectively and objectively the proposed method in terms of speed and effectiveness.

Future works can be a fast volume image segmentation approach by implementing the proposed method using a parallel device such as the graphics processing unit. One can also incorporate more kernels, i.e., more information in order to enhance the segmentation result.

REFERENCES

[1] Y. Chen, Z. Yan and Y. Chu, “Cellular automata based level set method for image segmentation”, IEEE/ICME International

Conference on Complex Medical Engineering, Beijing, pp. 23-27, May 2007.

[2] S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and beyond Numerical Mathematics and Scientific

Computation, Oxford University Press, New York, 2001.

[3] Ye Zhao, “Lattice Boltzmann Based PDE Solver on the GPU”, The Visual Computer, vol. 24, no. 5, pp. 323-333, Springer,

May 2007.

[4] X. He and L. Luo, “Lattice Boltzmann model for incompressible Navier-Stokes equation”, J. Stat. Phys., vol. 88, no. 3-4, pp.

927-944, 1997.

[5] P.L. Bhatnagar, E.P. Gross and M. Krook, “A model for collision processes in gases. I. Small amplitude processes in charged

and neutral one-component systems”, Phys. Rev., vol. 94, no. 3, pp. 511-525, 1954.

[6] W. Chen and Maryellen L. Giger, “A fuzzy c-means (FCM) based algorithm for intensity inhomogeneity correction and

segmentation of MR images”, IEEE International Symposium on Biomedical Imaging: Nano to Macro, vol. 2, pp. 1307-1310,

2004.

[7] J. Buick and C. Greated, “Gravity in a lattice Boltzmann model”, Phys. Rev. E, vol.61(5), pp. 5307–5320, 2000.

[8] X. Shan and G. Doolen, J. Stat. Phys., vol. 81, 379,1995.

 

Z. Wang 

CV 

Proposed   

Method 

 

16  

Test images Chan and Vese’s method Z. Wang et al.’s method Proposed method Human segmentation

1

2

3

4

5

6

7

8

9

10

17  

Fig.8. Segmentation results of the Chan- Vese’s method, the Z. Wang et al.’s method, the proposed method and manual segmentation.

[9] I. Ginzbourg and P.M. Adler, J. Phys. II, vol. 4, 191,1994.

[10] S. Chapmann and T.G. Cowling, The Mathematical Theory of Non-Uniform Gases: An account of the kinetic Theory of

Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge University Press, Cambridge, U.K., 1990.

[11] L. Chen, M. Lu and C.Philip Chen, “Multiple Kernel Fuzzy C-means based Image Segmentation”, Proceedings of IEEE

International Conference on Systems Man and Cybernetics (SMC), pp. 4123-4129, October 2010.

[12] D. Graves and W. Pedrycz, "Kernel-based fuzzy clustering and fuzzy clustering: a comparative experimental study," Fuzzy

Sets Syst., vol. 161, no. 4, pp. 522-543, 2010.

[13] C. Brun, N. Leporé, X. Pennec, Y. Chou, A. Lee et al., “A nonconservative Lagrangian framework for statistical fluid

registration-SAFIRA”, IEEE Transactions on Medical Imaging, vol. 30, no. 2, pp. 184-202 , 2011.

[14] A. Hagan and Y. Zhao, “Parallel 3D image segmentation of large data set on a GPU cluster”,G. Bebis et al. (Eds.): ISVC 2009,

Part II, LNCS 5876, pp. 960-969, 2009, Springer-Verlag Berlin Heidelberg 2009.

[15] S. Balla-Arabé, B. Wang and X.-B Gao, “Level set region based image segmentation using lattice Boltzmann method”,

Proceedings of the Seventh International Conference on Computational Intelligence and Security, pp. 1159-1163, Sanya,

China, December 2011.

[16] Z. Wang, Z. Yan and G. Chen, “Lattice Boltzmann method of active contour for image segmentation”, Proceedings of the

Sixth International Conference on Image and Graphics (ICIG), pp. 338-343, 2011.

[17] G. Papandreou and P. Maragos, "Multigrid geometric active contour models”, IEEE Transactions on Image Processing, vol.

16, no. 1, pp. 229-240, 2007.

[18] S. Osher and J. Sethian, “Fronts propagating and curvature dependent speed: algorithms based on Hamilton-Jacobi

formulation”, Journal of Computational Physics, vol. 79, no. 1, pp. 12-49, Nov. 1988.

[19] R. C. Gonzalez, R. E. Woods and S. L. Eddins, “Digital image processing using MATLAB”, Upper Saddle River, NJ:

Pearson/Prentice Hall, 2004.

[20] D. Zhang and S. Chen, “A novel kernelized fuzzy c-means algorithm with application in medical image segmentation”,

Artificial Intelligence in Medecin, Vol. 32, no. 1, pp. 37-50, September 2004.

[21] S. Osher and J. Sethian, “Fronts propagating and curvature dependent speed: algorithms based on Hamilton-Jacobi

formulations”, Journal of Computational Physics, vol. 79, no. 1, pp. 12-48, 1988

[22] M. Kass, A. Witkin and D. Terzopoulos, “Snakes: Active contour models”, International Journal of Computer Vision, vol.1,

no.4, pp. 321-331, 1988.

[23] V. Casselles, F. Catté and F. Dibos, “A geometric model for active contours in image processing”, Numerishe Mathematik,

vol. 66, pp. 1-31, 1993.

[24] R. Malladi, J. Sethian and B. Vemuri, “A topology independent shape modeling scheme”, Proceedings of SPIE Conference

on Geometric Methods in Computer Vision II, vol. 2031, San Diego, CA, pp. 246-258, 1993.

[25] V. Caselles, R. Kimmel and G. Sapiro, “On geodesic active contours”, International Journal of Computer Vision, vol.22, no

1, pp. 61-79, 1997.

[26] L. D. Cohen, “On active contour models and balloons”, Computer Vision, Graphics, and Image Processing., vol. 53, no.2,

pp. 211–218, 1991

11

18  

[27] N. Paragios and R. Deriche, “Geodesic active contours for supervised texture segmentation”, Proceedings of IEEE

Conference in Computer Vision and Pattern Recognition (CVPR), pp. I: 1034-1040, 1999.

[28] R. Ronfard, “Region based strategies for active contour models.,” International Journal of Computer Vision, vol.. 13, no. 2,

pp. 229–251, October 1994.

[29] T. Chan, L. Vese, Active contours without edges, IEEE Transactions on Image Processing, vol.10, no.2, pp. 266-277, 2001.

[30] X.-B Gao, B. Wang, D. Tao and X. Li, “A relay level set method for automatic image segmentation”, IEEE Transactions on

Systems, Man and Cybernetics Part B: Cybernetics, vol. 41, no. 2, pp. 518-525, April 2011.

[31] B. Wang, X.-B Gao, D. Tao and X. Li, “A unified tensor level set for image segmentation”, IEEE Transactions on Systems,

Man and Cybernetics Part B: Cybernetics, vol. 40, no. 3, pp. 857-867, 2010.

[32] D. Mumford and J. Shah, “Optimal approximations by piecewise smooth functions and associated variational problems”,

Commun. Pure Appl. Math., vol. 42, no. 5, pp. 577-685, 1989.

[33] S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer, 2003.

[34] S. C. Chen and D. Q. Zhang, "Robust image segmentation using FCM with spatial constraints based on new kernel-induced

distance measure", IEEE Transactions on Systems Man and Cybernetics Part B-Cybernetics, vol. 34, no. 4, pp. 1907-1916,

Aug, 2004.

[35] S. Zhu and A. Yuille, “Region competition: unifying snakes, region growing, and Bayes/MDL for multiband image

segmentation”, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 18, no. 9, pp. 884-900, 1996.

[36] K. Zhang, L. Zhang, H. Song and W. Zhou, “Active contours with selective local or global segmentation: a new formulation

and level set method”, Image and Vision Computing, vol. 28, issue 4, pp. 668-676, April 2010.

[37] S. Esedoglu, and Y. Tsai, “Threshold dynamics for the piecewise constant Mumford-Shah functional”, Journal of

Computational Physics, vol. 211, pp. 367-384, 2006.

[38] C. Li, C. Kao, J. Gore, and Z. Ding, “Implicit active contours driven by local binary fitting energy”, Proceedings of IEEE

Conference in Computer Vision and Pattern Recognition, pp. 1–7, 2007.

[39] C. Li, C. Xu, C. Gui, and M. D. Fox, “Level set evolution without re-initialization: a new variational formulation”,

Proceedings of IEEE Conference in Computer Vision and Pattern Recognition, vol. 1, pp. 430–436, 2005.

[40] J.C. Bezdek, “Pattern recognition with fuzzy objective function algorithms”, New York: Plenum Press, 1981.

[41] K. Charansiriphaisan, S. Chiewchanwattana and K. Sunat, “The expelling Gaussian-base kernel fuzzy c-means algorithm”,

Proceedings of the Third International Conference on Knowledge and Smart Technologies, pp. 44-50, July 8-10, 2011.

[42] B. Zhong, H. Yao and S. Liu, “Robust background modeling via standard variance feature”, Proceedings of ICASSP, pp.

1182-1185, 2010.

[43] C. Li, C. Xu, C. Gui and M.Fox, “Distance regularized level set evolution and its application to image segmentation”, IEEE

Transactions on Image Processing, vol. 19, no. 12, pp. 3243-3254, December 2010.

[44] M. Bauchemin, K. Thomson and G. Edwards, “On the Hausdorff distance used for the evaluation of segmentattion results”,

Canadian Journal of Remote Sensing, vol. 24, no. 1, pp. 3-8, 1998.

[45] S. Chabrier, H. Laurent, C. Rosenberger and B. Emile, “Comparative study of contour detection evaluation criteria based on

dissimilarity measures”, EURASIP Journal on Image and Video Processing, Volume 2008, Article ID 693053, 13 pages,

2008.

[46] D. Martin, C. Fowlkes, D. Tal and J. Malik, “A database of human segmented natural images and its application to

evaluating segmentation algorithms and measuring ecological statistics”, Proceedings of the 8th International Conference

in Computer Vision, vol. 2, pp 416-423, July 2001.

[47] U. Frisch, D. d’Humieres, B. Hasslacher, P. Lallemand and Y. Pomeau, Complex Syst. 1, 649 (1987).

19  

[48] S. Balla-Arabe´and X.Gao, “Image multi-thresholding by combining the lattice Boltzmann model and a localized level set

algorithm”, Neurocomputing, vol.93, pp. 106–114, 2012.

[49] N. Paragios and R. Dericheb, “Geodesic active regions and level set methods for motion estimation and tracking”,

Computer Vision and Image Understanding, vol.97, pp. 259–282, 2005.

[50] A. Mansouri, A Mitiche and C. Vázquez, “Multiregion competition: A level set extension of region competition to multiple

region image partitioning”, Computer Vision and Image Understanding, vol.101, pp. 137–150, 2006.