A novel dual minimization based level set method for image segmentation

Author’s Accepted Manuscript A Novel Dual Minimization Based Level Set Method for Image Segmentation Hai Min, Xiao-Feng Wang, De-Shuang Huang, Wei Jia www.elsevier.com/locate/neucom

PII: DOI: Reference:

S0925-2312(16)30767-6 http://dx.doi.org/10.1016/j.neucom.2016.07.023 NEUCOM17379

To appear in: Neurocomputing Received date: 6 September 2015 Revised date: 7 June 2016 Accepted date: 4 July 2016 Cite this article as: Hai Min, Xiao-Feng Wang, De-Shuang Huang and Wei Jia, A Novel Dual Minimization Based Level Set Method for Image Segmentation, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.07.023 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

A Novel Dual Minimization Based Level Set Method for Image Segmentation Hai Mina,d, Xiao-Feng Wang b,*, De-Shuang Huang c , Wei Jia d a

Center of Medical Physics and Technology & Cancer Hospital, Hefei Institutes of physical science, Chinese Academy of Sciences, Hefei Anhui 230031, China

b

Key Lab of Network and Intelligent Information Processing, Department of Computer Science and Technology, Hefei University, Hefei Anhui 230601, China

c

Machine Learning and Systems Biology Laboratory, Tongji University, Shanghai, 201804, China d

School of Computer and information, Hefei University of Technology, Hefei, 230009, China

[email protected], [email protected], [email protected], [email protected] Abstract: In this paper, we propose a novel dual minimization (DM) method based on level set to segment images with intensity inhomogeneity. Considering the variance of intensity inhomogeneity, we introduce an energy term based on multi-layer structure and further incorporate it into so-called optimal evolution layer which is used to construct final energy functional. Specially, by optimizing each layer of energy term based on multi-layer structure, we obtain multiple intensity centers in local neighborhoods with different sizes of inside and outside of contour. Then, the multi-layer intensity differences are constructed by utilizing multiple intensity centers to describe each pixel point. Next, we use the proposed dual minimization method to incorporate and minimize the energy term based on multi-layer structure. On one hand, we obtain the optimal evolution layer by minimizing the multi-layer energy term. On the other hand, we obtain the final segmentation results by minimizing the final energy functional based on optimal evolution layer. The multi-layer structure extracts more intensity information and the dual minimization method adaptively determines the desirable local region size for each pixel so as to solve the problem of variance of intensity inhomogeneity. The partition of local regions in optimal evolution layer induces the accurate segmentation results. Experimental results and quantitative experimental comparisons demonstrate that the proposed method is more robust and accurate in segmenting images with intensity inhomogeneity than the classical LIC and LBF models. Keyword: image segmentation; level set; intensity inhomogeneity; dual minimization; multi-layer structure 1. Introduction Level set methods have been extensively applied to image segmentation [1-5]. The fundamental idea of level set method is to represent a contour as the zero level set of a higher dimensional function and formulate the motion of the contour as the evolution of the level set function. By minimizing the level set function, the evolving contour can be forced to approximate the object boundary in image. Existing level set models for image segmentation can be categorized into two major classes: edge-based models [6-10] and region-based models [1, 11-18]. Edge-based level set models consider boundary as a discontinuation in gray values. However, it is not easy to detect weak boundary in images. Besides, this type of method is sensitive to the placement of initial contour. To address the problem, Chan and Vese [1] proposed a region-based level set model, i.e. CV model, which incorporates the Mumford-Shah [11-12] functional into a level set framework to give a piecewise *

Corresponding author: Xiao-Feng Wang ([email protected]) 1

constant representation of image. Region-based models aim to identify each region of interest by using a certain region descriptor to guide the motion of the evolving contour. However, it is very difficult to define a region descriptor for images with intensity inhomogeneities. Most of region-based models are based on the assumption of intensity homogeneity. A typical example is piecewise constant (PC) models proposed in [1, 14-15]. In [16-17], piecewise smooth (PS) models were proposed based on a general piecewise smooth formulation originally proposed by Mumford and Shah [11]. These methods do not assume that image intensities are homogeneous, and therefore are able to segment images with intensity inhomogeneities. However, these methods are too computationally expensive and are quite sensitive to the initialization of the contour [13], which greatly limit their utilizations. Intensity inhomogeneity often occurs in real-world images due to various factors which is a challenge to the image segmentation. To segment images with intensity inhomogeneities, researchers have proposed many algorithms [18-20] that incorporated the local spatial information into the global region-based level set methods. Due to the space limitation, we here only review several recently proposed models which are closely related to the present work. For example, Darolti et al. proposed the local region descriptor (LRD) model [18] to tune a balloon force parameter for images with strongly overlapping intensity distribution, but how to define the degree of the overlap is not mentioned in [18]. Li et al. [19] proposed the local binary fitting energy functional by incorporating the local energy into the conventional piecewise smooth model. Due to the assumption of local piecewise smoothness, these models [18-19] based on local region find it challenging to produce accurate segmentation when images are corrupted by serious intensity inhomogeneity. Lankton et al. [22] proposed the localized region-based model (LRB) which evolves a contour based on local information. The LRB model has two drawbacks. First, the used Dirac functional is restricted to a neighborhood around the zero level set, which makes the level set evolution act locally. Therefore, the evolution can be easily trapped in local minima [1]. Second, the region descriptor in LRB model is only based on the region mean information with fixed scale and does not consider the region variance. This may lead to inaccurate segmentation. In [20], authors proposed a variational model for tissue classification of MR images. In their model, the local intensities of different tissues within a local region were used to form different clusters. This local clustering criterion was combined with membership functions to form the energy functional. Finally, the tissue classification is performed by estimating the membership function and parameters that approximate the true signals in each region. Li et al. [21] proposed local intensity clustering criterion to construct the local intensity clustering (LIC) model. The LIC model can be considered as a locally weighted K-means clustering method [21]. However, it does not consider the clustering variance, which may cause inaccurate segmentation. Similar drawback also exists in the K-means clustering based method [23]. Generally, intensity inhomogeneity is usually described as a smooth and spatially varying field multiplying by the constant true signal of the same object in the measured image [24]. As mentioned above, the traditional methods utilize local regions to approximate the intensity inhomogeneity [21] or look upon local regions as intensity homogeneity [19, 22]. However, how to select the size of local region and the desirable reasons are not be gave. In view of this, we firstly introduce an energy term based on multi-layer structure. Next, we optimize each layer energy term based on multi-layer structure to obtain intensity centers of local regions. Further, we use multiple intensity centers to approximate each pixel point so as to obtain the multi-layer intensity differences between multiple intensity centers and each pixel point. Then, the so-called dual minimization (DM) method is proposed to realize two goals: First, it determines the optimal evolution layer by using the first minimization in the energy term

2

based on multi-layer structure so that the desirable local region sizes are obtained. Meanwhile, the final energy functional is constructed by optimal evolution layer. Second, it obtains the final segmentation results by using the second minimization in the final energy functional. In this paper, we make two main contributions. Firstly, we introduce the multi-layer structure into the level set energy functional to derive the multiple intensity centers of local regions. So, more intensity information is extracted by computing the multi-layer intensity difference. Secondly, the dual minimization method is first proposed whose utility and effectiveness are demonstrated by segmenting images with intensity inhomogeneity. The main contribution behind the dual minimization method is that we derive the optimal evolution layer. The optimal evolution layer which adaptively determines the desirable local region size for each pixel can solve the problem of variance of intensity inhomogeneity. The remainder of this paper is organized as follows: In Section 2, we review the related LBF model and LIC model. The proposed method is then described in Section 3. In Section 4, the experiments are presented to validate the effectiveness and robustness of our method for segmenting images with intensity inhomogeneity. Section 5 give some experimental discussions. Finally, the conclusion is included in Section 6. 2. Background Knowledge 2.1 LBF Model To improve the performance of CV model, Li et.al [19] proposed the LBF model by embedding the local information into piecewise smooth model. The energy functional of LBF model is written as follows:

E LBF  1 

   

k ( x  y )( I ( y )  f1 ( x)) 2 dydx

 inside ( c ) 

k ( x  y )( I ( y )  f 2 ( x)) 2 dydx    l  vp( )

(1)

2  outside ( c ) 

where 1 , 2 are fixed parameters, I is an input image, k is the Gaussian kernel with standard deviation  ,  and v are two fixed parameters. The parameter  is a constant to control the size of local region. p( ) is used to avoid the re-initialization step. The local intensity information is introduced so as to enable LBF model to handle intensity inhomogeneity problem. However, due to the variances of different local regions in whole image, the intensity inhomogeneities cannot be described in local region with fixed size for the whole image. LBF model is generally unavailable for images with severe intensity inhomogeneity. 2.2 LIC Model Recently, Li et al. [21] incorporated the bias field estimation and a local intensity clustering criterion to address the images with intensity inhomogeneities. The energy functional is written as follows:

E LIC  1 

   

k ( x  y )( I ( y )  b( x)c1 ) 2 dydx

 inside ( c ) 

k ( x  y )( I ( y )  b( x)c2 ) 2 dydx    l  vp( )

(2)

2  outside ( c ) 

where b( x) approximates intensity inhomogeneities in the given image. According to clustering criterion introduced in [21], the approximation of the clustering center of the local region can be defined as b( x)c1 or b( x)c2 . The advantage of LIC model is that it can segment some images with 3

intensity inhomogeneity by using cluster center to describe each pixel. The disadvantage should be detailedly discussed in the next section. 3. Model Description 3.1 Problem Description The phenomenon of intensity inhomogeneity often occurs in obtained digital images. Generally, an image I ( x) containing the intensity inhomogeneity b( x) is approximated by the following model:

I ( x)  b( x) J ( x)  n( x)

(3)

where J ( x) is the clean image free of intensity inhomogeneity and n( x) is the noise. To simplify the problem, the noise is often ignored since it is independent of the intensity inhomogeneity. It can be assumed that the intensity inhomogeneity b( x) is slowly varying in the whole image domain. Based on the above assumption, Li et al. [21] proposed the local intensity clustering criterion. They define a circular neighborhood with radius  centered at each pixel y   ,  y

x : x  y    .

For a

slowly varying intensity inhomogeneity b , the following formula is satisfied:

b( x)  b( y) for x   y

(4)

If the image domain  can be represented by several disjoint regions:

j

 j   , b( x) J ( x) in each sub-region  y

n j 1

 j   and

 j can be approximated by the constant b( y )c j .

b( x) J ( x)  b( y)c j for x   y

j

(5)

Thus, the constant b( y )c j is seemed as the clustering center of I ( x) in the LIC model.

I ( x)  b( y)c j for x   y

j

(6)

However, there are two disadvantages associated with LIC model. Based on the local intensity clustering criterion, LIC model computed clustering centers by the local regions with predefined and unified scale. It should be noted that the LIC model may captures limited information due to its limited range [27]. Since the varied intensity inhomogeneities often occur in images, the variation degree of intensity inhomogeneity in the whole image cannot be determined and measured in local regions with the unified scale. Meanwhile, all the image details of pixel neighborhood can not be captured by local regions with the unified scale. Therefore, if the size of local region is simply viewed as an unified value, it is difficult to accurately determine the local intensity clustering center. The results are usually unsatisfactory. In images with severe intensity inhomogeneity, the intensity distributions of local regions with the same size may be different (as shown in Fig.1). In Fig.1 (a), two local regions with the same size are enclosed by two blue circles in the background region. The corresponding intensity histograms of two local regions are shown in Fig.1 (b). It is obvious that the intensity distributions of two histograms are totally different. In Fig.1 (c), two local regions with the same size are represented by two green circles in the object region. The corresponding intensity histograms are demonstrated in Fig.1 (d) where intensity distribution differences still exists. That is to say, it is not reliable to recover the intensity inhomogeneity of the whole image by only utilizing local regions with unified size. 4

9

14

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12

7 10 6 5

8

4

6

3 4 2 2

1 0

0

100

200

(a)

300

0

0

100

200

300

(b) 18

40

16

35

14

30

12 25 10 20 8 15 6 10

4

5

2 0

(c)

0

100

200

300

0

0

100

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(d)

Fig.1 Intensity distributions difference between different local regions: (a) Two local regions labeled in background region. (b) The intensity histograms of two local regions of (a). (c) Two local regions labeled in background region. (d) The intensity histograms of two local regions of (c).

Unfortunately, LIC model does not provide the solution of finding the desirable local region for each pixel. Thus, it is unavailable to capture the local variation of intensity inhomogeneity in image with severe intensity inhomogeneity. The desirable method is to use multi-scale techniques where local regions with different sizes are used to capture more information [28]. In other words, the incorporation of multiple local regions induces the partition of entire image domain. To solve the problem, we consider proposing a two-step scheme to determine all the desirable local regions and optimal energy functional so as to segment images with severe intensity inhomogeneity. First, we introduce the local regions based on multiple sizes to construct the intensity centers respectively for each pixel. Here, we used the intensity centers which are obtained in each size to approximate the each pixel so that the so-called multi-layer intensity differences are computed. This approximation process in each size is called one layer. Meanwhile, due to the introduction of multiple sizes, multiple layers are obtained by local regions with multiple sizes and we called it a multi-layer structure. Second, the dual minimization method is proposed to determine the optimal evolution layer and minimize the final energy functional. The optimal evolution layer is composed by desirable local regions with different sizes for each point. Finally, the partition of local regions in optimal evolution layer can induce the final image segmentation. 3.2 Multi-Layer Structure In this section, the multi-layer structure is proposed and incorporated into LIC model. In our approach, we firstly choose a set of local regions with different sizes. The multiple sized local regions can generate more local intensity information allowing us to obtain better local details. For each size, we optimize each layer over the corresponding local region to obtain the approximation intensity inhomogeneities and intensity centers. Then, an energy term based on multi-layer structure is

5

presented. Firstly, we use  i to denote the scale of Gaussian kernel function. The Gaussian kernel function is written as:

 1 u 2 i2 , u  i  e K i (u )   a  0, otherwise  2

i  1 , 2 m

(7)

Here, we consecutively apply a series of Gaussian kernel functions with different scales

 i (  i  4i  1 ) to the input image. i denotes the layer index. Thus, the local region size at i-th layer is defined by 4i  1 . The goal is to generate a scale-space representation of local intensity information. As shown in Fig.2, the green points denote the pixels of image. The three red rectangles separately denote local regions with different sizes centered in green point. The local regions with coarser (bigger) sizes are more informative and robust against noise due to the averaging effect of Gaussian kernel function. Meanwhile, the local regions with finer (smaller) sizes derive more details of intensity distribution [27]. Then, based on the local intensity clustering criterion [22], the multiple local regions are separately utilized to approximate different intensity centers.

Fig.2 The diagram of multiple local neighborhoods with different sizes.

Then, according to local intensity clustering criterion, we construct the energy term EM based on multi-layer structure:



(1) ( 2) EM  EM , EM

(i ) EM



(8)

EM(i )  1  k i ( x  y )  ( I ( y )  b(i ) ( x)c1( i ) )2  H ( )dydx  2  k i ( x  y )  ( I ( y )  b(i ) ( x)c2(i ) )2  (1  H ( ))dydx

i  1 , 2 m

(9)

where EM(i ) denote the i-th layer of energy term based on multi-layer structure, b(i ) denotes the approximated intensity inhomogeneity, c1( i ) and c2( i ) separately denote intensity constants of interior and exterior of contour . b(i ) ( x)c1(i ) and

b(i ) ( x)c2(i ) denote the intensity centers of local region in

interior and exterior of contour and are estimated in local region with scale  i . As given in [20],

H in the above is approximated by the following formula: 1 2 x H ( x)  [1  .arctan( )] 2  

(10)

The differential of H is written as:

 ( x)  H '( x) 

6



1

   x2 2

(11)

For fixed  and b(i ) ( b(i ) is initialized to constant matrix where all elements are 1), the optimal c1( i ) and c2( i ) can be obtained by minimizing energy EM(i ) ( , b(i ) ) :

(i ) 1

c



c2(i )  For fixed  , c1( i ) ( i  1, 2

and

 (b  ((b

(i )

 k i )  I  H ( )dy

)  k i )  I  H ( )dy

(i ) 2

 (b

(i )

i  1, 2

 k i )  I  (1  H ( ))dy

i  1, 2

(i ) 2  ((b )  ki )  I  (1  H ( ))dy

m

m

c2( i ) , the multiple intensity inhomogeneity approximations

(12)

(13)

b(i )

m ) are also derived by minimizing energy EM(i ) ( , c1(i ) , c2(i ) ) :

b( i ) 

k i  ( I (c1(i )  H ( )  c2(i )  (1  H ( )))) k i  ((c1(i ) )2  H ( )  (c2( i ) )2  (1  H ( )))

i  1, 2

m

(14)

Thus, the i-th layer of intensity difference between intensity center and each pixel is obtained as

D j (i ) (i  1, 2

m , j  1, 2) . D j (i ) ( x)=( I ( x)  b(i ) ( x)c(ji ) )2

x  ,

(15)

where j  1, 2 separately denotes the interior and exterior regions of evolving contour. Here, D j (i ) denotes the intensity differences between each pixel I ( x) and intensity center obtained by local region with scale  i . Finally, the multi-layer intensity difference D j is denoted as follows: (2) ( m) D j  D(1) j , Dj ， Dj 

(16)

3.3 Dual Minimization Method In order to adaptively determine desirable local region for each pixel, the multi-layer intensity difference should be transformed to a single-layer intensity difference. The obtained single-layer can be called the optimal evolution layer which is incorporated into level set so as to construct the energy functional. Then, the accurate intensity center should be approximated and the corresponding local region size can be adaptively determined. Here, we proposed a so-called dual minimization method to determine optimal evolution layer and accurate intensity center. The detail reasons are given in the next paragraph. According to the idea of clustering method and the definition of D1(i ) and D2(i ) , the desirable intensity center inside (outside) of contour should make the intensity difference D1(i ) ( D2(i ) ) smaller [25]. If the intensity difference between intensity center and pixel intensity is smaller, the intensity center shall be thought as the desirable one. To obtain the better intensity centers of local regions, the 7

first minimization between multiple layers should be computed. Based on intensity inhomogeneity difference description mentioned above in Fig.1, the intensity histograms of multiple local regions with the same size have different bandwidths. Meanwhile, the desirable intensity centers can properly describe each pixel point so as to reply the variance of intensity inhomogeneity. Therefore, it can be seen that the partition of desirable local regions can induce better image segmentation results than that of local region with a unified size. Meanwhile, according to level set theory, the smaller value of energy functional is defined, the better the segmentation shall be obtained. The second minimization can be used to minimize the optimal evolution layer. Therefore, we need to utilize dual minimization method for all x in x   . differences D j (i ) (i  1, 2

The dual minimization method where not only makes the intensity

m , j  1, 2) between multiple layers to be minimized but also makes

optimal evolution layer to be iteratively updated the intensity centers. Here, the construction process of the optimal evolution layer is graphically represented in Fig.3. To detect the optimal local region size for each pixel, the minimization of multiple intensity differences should be firstly performed according to (17).

L( x)  arg min( D j (i ) ( x))  arg min(( I ( x)  b(i ) ( x)c(ji ) ) 2 ) i

i  1 , 2 m

(17)

i

where L(x) denotes the founded layer index i at pixel x which has the minimum intensity differences D j (i ) ( x) . Then, the optimal local region size s(x) at pixel x can be computed based on (18):

s( x)  4L( x)  1

(18)

Finally, the optimal local region sizes of all pixels form the optimal evolution layer based on its position in the original image. So, the optimal evolution layer is actually the two dimensional set of optimal local region sizes. Multiple intensity difference Layer 1 Layer 2

Pixel 1

Optimal local region size

Minimization of intensity difference

Layer m Layer 1 Layer 2

Minimization of intensity difference

Optimal evolution layer Optimal local region size

Pixel n

Layer m Multiple intensity difference

Fig.3 The construction process of optimal evolution layer.

Further, we use Fig.4 to show the process of so-called dual minimization. The red rectangles denote

8

each intensity difference of each layer: D j (i ) (i  1, 2

m ; j  1, or, j  2) . Green rectangles denote

the intensity differences for the same pixel at each layer which needs to be minimized. Minimization with respect to the intensity differences between multiple layers is solved and called as the intensity differences z j of the optimal evolution layer as shown in the left of Fig.4. Then, minimization with respect to the intensity differences z j on the optimal evolution layer, i.e. contour evolution, is performed as shown in the right of Fig.4. D (1)j Minimization

Incorporation Minimization

zj

D (ji )

D (j m )

Fig.4 The diagram of dual minimization method

The intensity differences of optimal evolution layer are computed as follows:

z j ( x)  min ( D j (i ) ( x))  min (( I ( x)  b(i ) ( x)c(ji ) )2 ) i1,2 m

i1,2 m

(19)

Then, the intensity differences of optimal evolution layer are incorporated into the level set energy functional which is written as follows:

ED  1  z1 ( x)  H ( )dx  2  z2 ( x)  (1  H ( ))dx 



(20)

The dual minimization method has two main advantages. Firstly, the minimization of multi-layer intensity difference means that the desirable description or approximation for each pixel point is determined. Meanwhile, the optimal evolution layer is obtained. Secondly, unlike the LIC model, we propose the dual minimization method to determine intensity centers of local regions centered at each pixel point. Since the optimal intensity centers are adaptively determined, the proposed method can better solve the variance of intensity inhomogeneity and is more meaningful than LIC model. Meanwhile, it is not to find the local region with larger or smaller size, but to find desirable local region in which the intensity differences between multiple intensity centers and pixel are minimized. The each desirable local region can be effectively used to induce the accurate partition of image. To preserve the regularity of level set function and smooth the evolving contour, some re-initialization methods [20, 29-30] have been proposed. Here, we utilize the widely used regularization term ER proposed in [20].

ER    l  vp( )     H ( ) dx  v  

The final energy functional is defined as follows:

9



1 (  1)2 dx 2

(21)

E ( )  ED  ER  1 

min (( I ( x)  b (i ) ( x)c1(i ) ) 2 )  H ( )dx

 i1,2 m

 2 

min (( I ( x)  b (i ) ( x)c2(i ) ) 2 )  (1  H ( ))dx

 i1,2 m

    H ( ) dx  v  



(22)

1 (  1) 2 dx 2

In our method, the optimal number of layers m is determined by experimental experience. We have done a lot of experiments to test the influence of the number of layers on final segmentation. Finally, we found that 8 layers are suitable for segmenting image with size smaller than 400  400 . While the image size is larger than 400  400 , the optimal number of layers is found to be 16. In this paper, the sizes of all test images are smaller than 400  400 . So, the optimal number of layers m is set to 8. 3.4 Numerical Computation By minimizing energy functional E in (22), we obtain the image segmentation result given by the level set function  . The energy minimization is achieved by an iterative two-step process. At each iteration, we firstly minimize the description differences between multiple layers. Then, we minimize the energy E with respect to each of its variables  , b(i ) , c1( i ) and c2( i ) and give the solution to the energy functional minimization. For fixed b(i ) , c1( i ) and c2( i ) , the minimization of E ( , b(i ) , c1(i ) , c2(i ) ) with respect to  can be achieved by using standard gradient descent method:

 E  t 

(23)

Then, we express the corresponding gradient flow equation as:

   ( )( min ( D1(i ) )  min ( D2 (i ) )) i1,2 m i1,2 m t data term

    ( ).div( )  v( 2  div( )  

(24)

regularization term

Where  is gradient operator, div(.) is the divergence operator. Here, we shall summarize the whole algorithm steps:

 1 x  inside(c) Step 1: Initialize  0 (x)   , b(i ) ( x)  1 , n  0 ,  i  4i  1 ;  1 otherwise  Step 2: Compute c1( i ) , c2( i ) by virtue of formulas (12) and (13); Step 3: Compute b(i ) using formula (14); Step 4: Compute the minimization between multi-layer structures according to formula (17); Step 5: Minimize the energy functional E ( ) by using formula (24); Step6: Algorithm ends if iteration number is reached, or else initialize b(i ) ( x)  1 and return to step 2.

10

4. Experiments In this section, we shall present the experiments of the proposed dual minimization (DM) method and LBF and LIC models on some images with intensity inhomogeneity and then give quantitative comparison performance. The proposed method was implemented by Matlab 7 on a computer with Intel Core i7-3770 Duo 3.4GHz CPU, 8G RAM, and Windows 7 operating system. Parameter  of the proposed method is selected from three values: 0.0001 2552 , 0.001 2552 and 0.01 2552 in for all experiments in this paper. Fig.5 shows the segmentation results of the proposed DM methods for computed tomography angiography (CTA) images which includes two blood vessel images and one cell image. The curve evolution processes are depicted by showing the initial contours (in the left column), intermediate contours (in the middle column), and final contours (in the right column) on the images. The experiment shows that DM method is able to provide satisfying segmentation results for such images with intensity inhomogeneity.

Initial contour

20 iterations

78 iterations

Initial contour

16 iterations

44 iterations

Initial contour

8 iterations

16 iterations

Fig.5 The segmentation results of the proposed DM method for images with intensity inhomogeneity: The first column: initial contours. The second column: intermediate contours. The third column: final contours.

The piecewise Gaussian generalization and local region-based models have been the focus of most studies because of their tractability. In the following, the performance of DM method is demonstrated by comparing with related LBF model [20] and LIC model [22]. Similar to DM method, LBF model applied the Gaussian kernel to form the local region so as to describe the intensity in each local region. Besides, the local intensity clustering criterion proposed in LIC model [22] is used in DM method. To make a fair comparison, we select the optimal scale for the LBF and LIC model in all experiments. The following experiment shows that DM method is more robust against intensity inhomogeneity than the LIC model (as shown in Fig.6). In order to make fair comparison and testify the advantage of the proposed method, we provide the optimal initial contours and try using multiple scales for LIC model to obtain the optimal result. The blue curves of Fig.6 (a)-(g) denote the segmentation results of LIC 11

model with different scales i.e.  =2 ,  =3 ,  =5 ,  =8 ,  =10 ,  =13 and  =16 . It can be seen that LIC model failed to extract the object boundaries. Fig.6 (h) depicts the segmentation results of DM method which are more accurate. Due to the introduction of multi-layer structure and dual minimization method, the desirable local intensity clustering centers are implied and corresponding local regions are different in the whole image. Then, DM method can adaptively determine the desirable clustering center so as to be more reasonable than the LIC model. Thus, the corresponding local regions of desirable intensity centers are more suitable for image segmentation. 230 iterations

160 iterations

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230 iterations

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(c)

230 iterations

130 iterations

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130 iterations

160 iterations

(d) 300 iterations

500 iterations 160 iterations

(e)

(f)

(g)

(h)

Fig.6 The comparison of segmentation results between LIC model and DM method: (a) segmentation result of LIC model with scale  =2; (b) segmentation result of LIC model with scale  =3; (c) segmentation result of LIC model with scale  =5; (d) segmentation result of LIC model with scale  =8; (e) segmentation result of LIC model with scale  =10; (f) segmentation result of LIC model with scale  =13; (g) segmentation result of LIC model with scale  =16; (h) segmentation result of DM method.

The next experiment illustrates the robustness of DM method for segmenting three images with severe intensity inhomogeneity (as shown in Fig.7). We also compared DM method with LBF model and LIC model both of which posses certain capability of handling intensity inhomogeneity. To obtain best results, we still predefine optimal scales and initial contours for LBF and LIC models. The segmentation results of the LBF and LIC models are denoted by blue contours and separately shown in Fig.7 (a) and Fig.7 (b). The final segmentation results of DM method are denoted by red contours and shown in Fig.7 (c). Obviously, LBF and LIC models both obtained failed segmentation results while DM method successfully segmented three images. To further measure the quality of the extracted objects, we invited six individuals to manually select object boundaries from the images which are regarded as the ground truth. Here, Jaccard similarity coefficient is used as a quantitative measure to

12

evaluate the segmentation results of three methods.

J (Om , Ot ) 

A(Om A(Om

Ot ) Ot )

(25)

where Om denotes the derived object region by the algorithm and Ot denotes the corresponding object region in the ground truth image. A() represents the area of region. It is noticed that the Jaccard similarity coefficient is bounded in [0, 1] and larger value implies more accurate segmentation. The corresponding Jaccard similarity coefficients of three methods for Fig. 7 are shown in Table 1. It can be seen that DM method is superior in term of accuracy than LBF model and LIC model. Due to the local variation of intensity inhomogeneity, LBF model can not accurately describe all the local regions with fixed scales and LIC model can not determine desirable clustering centers for all pixel points. So, they failed to segment images of Fig.7. Final contour, 500 iterations

500 iterations

130 iterations

360 iterations Final contour, 800 iterations

160 iterations Final contour, 800 iterations

100 iterations

(a)

(b)

(c)

Fig.7. The segmentation results of LBF model, LIC model and DM method on images with severe intensity inhomogeneity. (a) Segmentation results of LBF model. (b) Segmentation results of LIC model. (c) Final segmentation results of DM method. Table 1. Jaccard similarity coefficients of LBF model, LIC model and DM method for images in Fig.7. Method

Image1

Image2

Image3

LBF LIC

0.8879 0.5424

0.8968 0.3955

0.8531 0.6696

DM

0.9967

0.9793

0.9916

To further testify the superiorities of the proposed DM method, we gave the comparison experiments between LBF model, LIC model and DM method on real images. Fig.8 shows three medical images including bone, brain tumor, and tumor images. All the images contain severe intensity inhomogeneity which are difficult for traditional models. Here, the optimal scales and initial contours are still provided 13

for the LBF and LIC models. The segmentation results of LBF model, LIC model and DM method can be found in Fig.8 (a), (b) and (c). The proposed DM method can not only minimize the local energy with different scales for each pixel of level set function but also construct the optimal evolution layer (inside and outside of contour). However, the LBF model only partitions the image in local region with fixed size and LIC model cannot determine the desirable local intensity clustering center according to the variance of intensity inhomogeneity. Thus, the LBF and LIC models cannot obtain accurate segmentation results for severe intensity inhomogeneity images. The Jaccard similarity coefficients of three methods are listed in Table 2 which shows that DM method achieves better segmentation performance than LBF model and LIC model. 160 iterations

(a)

300 iterations

122 iterations

(b)

(c)

Fig.8. The segmentation results of LBF model, LIC model and DM method on real images with severe intensity inhomogeneity. (a) Segmentation results of LBF model. (b) Segmentation results of LIC model. (c) Segmentation results of DM method. Table 2. Jaccard similarity coefficients of LBF model, LIC model and DM method for images in Fig.8. Method

Image1

Image2

Image3

LBF LIC

0.8262 0.7189

0.5963 0.9679

0.9709 0.7411

DM

0.9970

0.9809

0.9916

In Fig.9, we give more experiment results to testify the superiority of DM method. The experiment includes three photograph images, an infrared image and two medical images. It can be seen that the complicated backgrounds are emerged in three photograph images. The object of infrared image is disturbed by intensity inhomogeneity. The white matter often suffers the intensity inhomogeneity and narrow boundaries. The heart image is intensity inhomogeneity. However, DM method can still accurately segment all images (as red contours shown in Fig.9). The segmentation results of both LBF model (as first column shown in Fig.9) and LIC model (the second column in Fig.9) are obviously failed. 14

600 iterations

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130 iterations

56 iterations

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130 iterations

70 iterations

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100 iterations

300 iterations

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10 iterations

600 iterations

130 iterations

9 iterations

Fig.9. The segmentation results on real images with severe intensity inhomogeneity. (a) Segmentation results of LBF model. (b) Segmentation results of LIC model. (c) Final segmentation results of DM method.

Besides, to demonstrate the robustness of the proposed DM method for different initial contours, we also compare DM method with the LIC model on three images (as shown in Fig.10). Here, we provide six different initial contours for each image which are denoted by the green contours. The blue and red contours separately denote the final segmentation results of LIC model and DM method. The corresponding Jaccard similarity coefficients are summarized in Table 3. It can be seen that all the Jaccard similarity coefficients of DM method are close or equal to 1. Despite the great difference among these initial contours, the segmentation results of DM method are almost same. On the contrary, the LIC model cannot segment three images accurately by using the first five initial contours. The multi-layer structure in DM method is used to capture more information and can not be constrained by local region. So, more boundaries information can be extracted by DM method. 15

280 iterations

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700 iterations

300 iterations

300 iterations

100 iterations

100 iterations

(b) Fig.10 The performance comparison of LIC model and DM method with different contour initializations. (a) The initial contours and final contours of LIC model. (b) The initial contours and final contours of DM method. Table 3. Jaccard similarity coefficients of LIC model and DM method by using different initial contours for images in Fig.10. Method LIC

DM

Column1

Column2

Column3

Column4

Column5

Row1 Row2 Row3

0.7428 0.5264 0.8126

0.7231 0.5622 0.7050

0.7369 0.5410 0.7764

0.7526 0.5893 0.5634

0.6231 0.5700 0.5235

Row1 Row2

0.9988 0.9963

0.9988 1

0.9988 0.9963

1 0.9963

0.9988 0.9963

Row3

0.9983

0.9964

0.9983

0.9991

0.9983

16

Column6 0.9963 0.9963 0.9972 1 1 0.9991

Here, we shall give the reason of initialization robustness of DM method. The purpose of using multiple layers is to build the optimal evolution layer and capture more local intensity information. As discussed above, the adaptively determined optimal local region at each pixel in optimal evolution layer is a region with minimum intensity difference. Therefore, the estimated intensities in optimal local regions can approximate the real intensity distribution or intensity center. Further, we gave the so-called scale maps in Fig.11 to demonstrate the optimal evolution layer. Here, different layer index i has different colors in scale map. So, the optimal local region size at each pixel is  i  4i  1 whose value can also be represented by color. It can be seen from the scale maps and corresponding segmentation results that the local region sizes of pixels near boundary are generally small. This is because that the smaller local region sizes can be used to capture detailed intensity information so as to detect boundary. Meanwhile, the pixels far from the boundary have larger local region sizes which is capable of detecting global contrast information so as to avoid being trapped in local minima. In addition, the pixels in regions near the image margin appear to have small local region sizes. Obviously, these regions have homogeneous intensity distributions and the small local region size can guarantee evolving contour not to be trapped in local minima. In traditional methods, the local region size is fixed in the whole image domain which makes these methods suffer from more or less over-segmentation. By constructing the optimal evolution layer which is beneficial to detect right boundary and constrain local minima, the proposed method is more robust to initialization than traditional methods. 8

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Fig.11 The scale maps and segmentation results. The first row: Scale maps. The second row: Segmentation results.

5. Discussions The distinctive contributions of our method compared to existing literatures are summarized as following: Due to the variance of local region intensity, it is difficult for traditional level set methods to segment images with severe intensity inhomogeneity. To solve this problem, the proposed dual minimization method adaptively determines the optimal evolution layer for each pixel. Further, the optimal local region size for each pixel can be determined so as to improve the segmentation capability for images with severe intensity inhomogeneity. In addition, the local intensity centers are derived at local regions with different sizes. The smaller local region sizes can be used to capture detailed intensity information and detect boundary. The pixels far from the boundary have larger local region sizes which is capable of detecting global contrast information and avoiding being trapped in local minima. Therefore, the proposed method can obtain robustness and stability segmentation results for different initializations and parameters.

17

5.1 Regularization term Further, we also used the reaction diffusion (RD) regularization method proposed in [31] to avoid the re-initialization. Here, we use RD-DM to denote the dual minimization method based on reaction diffusion and DM to denote the dual minimization method using regularization term in (21). The experimental comparisons of RD-DM and DM methods were illustrated in Fig.12. The blue curves in the first row denote the segmentation results of RD-DM method. The corresponding final level set functions are shown in the second row. The red curves in the third row denote the segmentation results of DM method and final level set functions are shown in the last row. Since the proposed method can adaptively determine optimal evolution layer so as to avoid the intensity inhomogeneity influence, the obtained segmentation results of RD-DM method are same with that of DM method. The corresponding iteration times are almost same. However, due to the advantages of anti-leakage and anti-noise of RD regularization method, the final level set functions of RD-DM method seem to be more smooth that that of DM method.

160 iterations

60 iterations

160 iterations

60 iterations

60 iterations

20 iterations

260 iterations

20 iterations

Fig.12 The experimental comparisons of RD-DM and DM methods. The first row: The segmentation results of RD-DM method. The second row: Final level set functions of RD-DM method. The third row: The segmentation results of DM method. The fourth row: Final level set functions of DM method.

5.2 Accelerated algorithm In this section, we shall firstly analyze the computation complexity of the proposed DM method and LIC model and then give our accelerated solution. Since the optimal number of layers m is set to 8 by

18

experimental experience, DM method has an inner loop whose iteration times is 8. The computation complexity of DM method is O(8n) . Comparably, LIC model does not need inner loop, so its computation complexity is O(n) . The computation complexity of DM method seems to be larger than that of LIC model. However, LIC model needs much more iteration times to segment an image. Here, we gave an experiment to show the segmentation performance of LIC model and DM method in the first and second rows of Fig.13. To make a fair comparison, we provided desired initial contours for LIC model. The corresponding iteration times and processing time of two methods are shown in the first and second rows of Table 4. It can be seen that LIC model spent much more iteration times to segment three images than DM method. The reason is due to that DM method computes a optimal evolution layer at each iteration so as to improve the contour evolving ability and reduce the iteration times. Although LIC model spent much more iteration times, its processing time was similar to that of DM method. It should be noted that LIC model adopts an acceleration algorithm to reduce the computation in the code. The estimation of intensity inhomogeneity b and two constants c1 , c2 are performed every 10 iterations. Totally speaking, the computation time of LIC model at each iteration is smaller than that of DM method.

20 iterations

32 iterations

44 iterations

13 iterations

20 iterations

10 iterations

Fig.13 Segmentation comparisons of LIC model, DM method and accelerated DM method. The first row: Segmentation of LIC model. The second row: Segmentation of dual minimization method. The third row: Segmentation of accelerated dual minimization method.

Table 4. Iteration times and processing time between LIC model, Dual minimization (DM) method and accelerated dual minimization (ADM) method on segmenting images in Fig.13. Fig.13 Image1:176*177

Fig.13 Image2: 88*85

Fig.13 Image3: 137*193

Iterations

Time (s)

Iterations

Time (s)

Iterations

Time (s)

LIC DM

300 20

1.82 1.77

300 30

1.19 1.17

360 44

2.60 3.22

ADM

30

0.75

100

0.49

80

0.96

Actually, the inner loop of computing the optimal evolution layer at each iteration is the main factor of time consuming. However, we found that the differences of the optimal evolution layer at each 19

iteration are generally small. It means that the updating of the optimal evolution layer needs not to be performed at every iteration. Here, we gave our acceleration solution which is to update the optimal evolution layer every k iterations. As a result, the processing efficiency of the dual minimization method is greatly enhanced. Correspondingly, the fifth step of algorithm steps can be revised as the following: Step 5: Minimize the energy functional E ( ) by using formula (24) and update the optimal evolution layer every k iterations. Generally, k can be set to 10 to obtain a satisfying result in our experiments. The segmentation results of the accelerated dual minimization (ADM) method are shown in the third row of Fig.13. The corresponding iteration times and processing time are shown in the last row of Table 4. Obviously, ADM method spent least processing time to obtain the same segmentation results. It can be said that the accelerated solution can not only improve the computation efficiency but also ensure the segmentation accuracy. 5.3 Experimental rigorousness In order to give fair comparison results, we have continuously adjusted the parameters and initial contours for LBF and LIC models in all of our experiments. In this section, we take the LIC model as an example and give two experiments to testify the rigorousness of our experiment. For the segmentation of the first image of Fig.6 by using LIC model, we have used different initial contours and adjusted regularization parameter  from 0.0001 to 0.1 with interval as 0.0001, local region size

 from 0.5 to 30 with interval as 0.5. In Fig.14 (a), we show parts of segmentation results (blue contours) and initial contours (green contours). It can be seen that LIC method still can not obtain good segmentation results even different parameters and initial contours were provided. For the second test image shown in Fig.14 (b), the LIC model can obtain correct segmentation result only when   2 ,   0.0001 and the initial contour is placed in right and bottom side of image (as shown in the first image in Fig.14(b)). Furthermore, we used the parameter   2 ,   0.0001 and continually change the initial contour. Unfortunately, LIC model could not segment the images in most of cases (as shown in the other images in Fig.14(b)).

(a)

20

(b) Fig.14 Segmentation results of LIC model with different initial contours and parameters. (a) Row1:   1 , Row2:   3 , Row3:   8 , Row4:   16 ; Column1:   0.0001 , Column2:   0.001 , Column3:   0.01 , Column4:   0.0001 , Column5:   0.001 , Column6:   0.01 , Column7:   0.0001 , Column8:   0.001 , Column9:   0.01 .

(b)   2   0.0001 ．

5.4 Drawbacks By using the dual minimization method to derive the optimal evolution layer, our method can obtain better local intensity centers than that of LIC model. The dual minimization method actually includes two minimizations. One is to solve the minimization between multiple layers which equals to the inner minimization of different local regions centered at each pixel. The other one is to determine the minimization of energy functional which discriminates each pixel into the object and background regions. Recently, a locally statistical active contour (LSAC) model [32] considering the intensity variance information has been proposed. By transforming the intensity into local region statistical information, the intensity overlay in the object and background regions is avoided. As a result, LSAC model can yield better segmentation results than LIC model. Here, we tried to extend our dual minimization method to the LSAC model to obtain better performance. Some experimental results were shown in Fig.15. It can be seen that the first two images were accurately segmented and the second two images could not be segmented. Noted that the original LSAC model can well segment all these images. The main reasons can be explained as follows: The LSAC model is based on maximum probability distribution and highlights the statistical feature which is denoted by Gaussian distribution. The Gaussian distribution information includes not only description difference between pixel and mean but also the variance information, which are all important to the final solution of the maximum probability distribution. However, we found that the dual minimization only works for description difference. For the failed segmentations, we analyzed the inner minimization. Smaller description differences and bigger variances were obtained. Smaller description differences mean that local regions are approximately intensity homogeneous. On the contrary, the bigger variances denote that local regions are intensity inhomogeneous Thus, the local region determined by dual minimization method becomes invalid in LSAC model. If the dual minimization is used in LSAC model, the maximum probability distribution of different local regions centered at each pixel is unavailable to classify the pixel into object or background region. 300 iterations

300 iterations

300 iterations

300 iterations

Fig.15 Segmentation results of LSAC model based on dual minimization The dual minimization method is proposed as a general framework which is used to improve the local region-based methods. Except the local region-based methods, edge-based methods have also been widely used in recent years. For example, Song [33] proposed regularizing the image Laplacian with an anisotropic diffusion term to reduce noise and preserve the object boundary. Here, we also tried

21

to extend the dual minimization idea to this method but failed. The purpose of dual minimization is to find optimal local region in which the intensity differences between multiple intensity centers and pixel are minimized. However, the minimization of image Laplacian based on multiple layers can derive Laplacian operation with big local region size near the boundary. As a result, the false boundary will be detected. In other words, the dual minimization method cannot act as a classifier of object and background pixels for model in [33]. In the future, we plan to incorporate the adaptive scale gradient operator into our dual minimization method. The operator can determine the scale of edge operator by analyzing the different local region information. So, the dual minimization maybe extended to the edge-based method. 6. Conclusion In this paper, the novel dual minimization method has been proposed to segment images with severe intensity inhomogeneity. The multi-layer structure has been introduced to capture more local intensity information and approximate multiple intensity centers. The energy term based on multi-layer structure has been constructed. Further, the dual minimization method is proposed to determine the optimal evolution layer and obtain the accurate segmentation results. Here, the accurate segmentation is induced by the partition of the local regions of optimal evolution layer. Applications of our method to images with intensity inhomogeneity show that it is more powerful to segment image objects than some classical models (such as the LIC and LBF models). The comparison experiments demonstrate that the proposed method can achieve superior segmentation results for images with severe intensity inhomogeneity. Meanwhile, the performance of the proposed method has also testified the robustness for the different initial contours of level set. Acknowledgements. This work was supported by the grant of the National Natural Science Foundation of China, Nos. 61005010, 61175022, 61402018, 61305006, the grants of Anhui Provincial Natural Science Foundation, Nos. 1308085MF84, 1508085QF116, the grant of Support Project for Excellent Young Talent in College of Anhui Province (X.F. Wang), the grant of Key Constructive Discipline Project of Hefei University, No. 2016xk05, the grant of Training Object Project for Academic Leader of Hefei University, No. 2014dtr08. References [1] T. Chan, L. Vese, “Active contours without edges,” IEEE Trans. Image Process., vol. 10, no. 2, pp. 266–277, 2001. [2] L. Cohen and I. Cohen, “Finite-element methods for active contour models and balloons for 2-D and 3-D images,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 15, no. 11, pp. 1131–1147, 1993. [3] Z. Hou, “A review on MRI image intensity inhomogeneity correction,” Int. J. Biomed. Imag., vol.2006, pp.1-11, 2006. [4] M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: active contour models,” Int. J. Comput. Vis, vol. 1, no.4, pp. 321–331, 1988. [5] B. Wang, X. Gao, J. Li, X. Li, D. Tao, “A level set method with shape priors by using locality preserving projections,” Neurocomputing, vol. 170, pp.188-200, 2015. [6] V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic active contours,” Int. J. Comput. Vis., vol. 22, no. 1, pp. 61–79, 1997. [7] S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum, and A. Yezzi, “Gradient flows and geometric active contour models,” in Proc. 5th Int. Conf. Comput. Vis., pp. 810–815, 1995. [8] R. Kimmel, A. Amir, and A. Bruckstein, “Finding shortest paths on surfaces using level set 22

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Bibliography of authors Hai Min received the B.E. degree from department of automation, Qing Dao University, Qingdao, China, in 2007, the M.Sc. and Ph.D degrees in Pattern Recognition and Intelligent System from University of Science and Technology of China, Hefei, China, separately in 2010 and 2014. His research interests include pattern recognition and image segmentation. Xiao-Feng Wang received the B.Sc. degree in Computer and Science Technology from Anhui University, Hefei, China, in 1999, the M.Sc. degree in Pattern Recognition and Intelligent System from Institute of Intelligent Machines, Graduate University of Chinese Academy of Sciences, Hefei, China, in 2005, the Ph.D degree in Pattern Recognition and Intelligent System from University of Science and Technology of China, Hefei, China, in 2009. He is now the dean of department of computer science and technology of Hefei University. His research interests include image processing, pattern recognition and data mining. De-Shuang Huang received the B.Sc., M.Sc. and Ph.D. degrees all in Electronic Engineering from Institute of Electronic Engineering, Hefei, China, National Defense University of Science and Technology, Changsha, China, and Xidian University, Xi'an, China, in 1986, 1989 and 1993, respectively. His research interests include pattern recognition, image processing and data mining. Wei Jia received the B.Sc. degree in informatics from Central China Normal University, Wuhan, China, in 1998, the M.Sc. degree in computer science from Hefei University of Technology, Hefei, China, in 2004, and the Ph.D. degree in pattern recognition and intelligence system from University of Science and Technology of China, Hefei, China, in 2008.His research interests include biometrics, pattern recognition, and image processing.

Hai Min

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Xiao-Feng Wang

De-Shuang Huang

Wei Jia

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