Robust level set image segmentation via a local correntropy-based K-means clustering

Pattern Recognition 47 (2014) 1917–1925

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Pattern Recognition journal homepage: www.elsevier.com/locate/pr

Robust level set image segmentation via a local correntropy-based K-means clustering Lingfeng Wang n, Chunhong Pan NLPR, Institute of Automation, Chinese Academy of Sciences, Beijing, 100190, China

art ic l e i nf o

a b s t r a c t

Article history: Received 7 August 2013 Received in revised form 5 November 2013 Accepted 13 November 2013 Available online 21 November 2013

It is still a challenging task to segment real-world images, since they are often distorted by unknown noise and intensity inhomogeneity. To address these problems, we propose a novel segmentation algorithm via a local correntropy-based K-means (LCK) clustering. Due to the correntropy criterion, the clustering algorithm can decrease the weights of the samples that are away from their clusters. As a result, LCK based clustering algorithm can be robust to the outliers. The proposed LCK clustering algorithm is incorporated into the region-based level set segmentation framework. The iteratively re-weighted algorithm is used to solve the LCK based level set segmentation method. Extensive experiments on synthetic and real images are provided to evaluate our method, showing significant improvements on both noise sensitivity and segmentation accuracy, as compared with the state-of-the-art approaches. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Image segmentation Level set Correntropy-based K-means

1. Introduction Image segmentation plays an important role in both image and video based pattern recognition systems. However, it is still a challenging problem since images are often distorted by complex noise and intensity inhomogeneity. In the past decades, a number of researchers have been attracted to solve these difficulties and have proposed many effective methods. Among them, level set based methods have proved to be a successful branch. The early level set based segmentation methods [1–9] utilize edge information to attract active contours toward the object boundaries. However, these methods are sensitive to noise. To solve this problem, region-based methods are proposed in [10–28]. Different from edge-based methods, region-based methods exploit the statistical information inside and outside the contour to guide the contour evolution. Hence, these methods are not sensitive to image noise and weak edges. One of the most famous region-based method is the CV model proposed by Chan et al. [10]. This method formulates the image segmentation problem as a K-means clustering. As pointed out in [19], as a global method, the CV model cannot solve intensity inhomogeneity well. To tackle this difficulty, many local methods have been proposed recently. A popular one is the local binary fitting (LBF), which is proposed by Li et al. [19]. The core idea behind the LBF is that it considers the K-means clustering problem locally. Motivated by [19], the local image fitting (LIF) and the local Gaussian fitting (LGF) have been presented in [21,29], respectively.

n

Corresponding author. Tel.: þ 86 10 62612582. E-mail addresses: [email protected] (L. Wang), [email protected] (C. Pan).

0031-3203/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.patcog.2013.11.014

Wang et al. [30] point out that the LBF method is sensitive to initialization. To prevent this, they propose a local order method in [30] (LBF þOrder). Besides improving global methods into local versions, many researchers focus on the convexity of segmentation models. They present many convex region-based methods [12,31,32]. For example, in [12], Nikolova et al. propose the Rudin–Osher–Fatemi total variation based CV model. In the above level set segmentation models, image noise is always assumed to be Gaussian. In practice, for many types of images, such as synthetic aperture radar (SAR) images, it is insufficient to assume that the noise is Gaussian. To make a segmentation algorithm still work under unknown complex noise, we propose a new segmentation method via the local correntropybased K-means (LCK) clustering. The main advantages of our segmentation method can be highlighted as follows. 1. Due to the correntropy criterion, the clustering algorithm adaptively increases the weights of samples that are close to their clusters, and decreases the weights of samples that are away from their clusters. As a result, our proposed LCK based segmentation algorithm is robust to outliers. 2. In the experiments on synthetic images, our method is more robust than the state-of-the-art approaches. Especially, in our results, foreground and background cluster images preserve edge features well even when the noise is large. In the experiments on real images, our method can provide accurate segmentation results even when the object and the background are very similar. The rest of this paper is organized as follows. Section 2 introduces the correntropy-based K-means clustering algorithm. Section 3 describes the proposed segmentation method. The optimization is

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L. Wang, C. Pan / Pattern Recognition 47 (2014) 1917–1925

Fig. 1. Comparisons of global and local methods. In subfigures (b, c), the rectangle is the desired segmentation contour, which segments image into two parts, i.e., foreground and background. The circle is a local region.

described in Section 4. Experimental results are provided in Section 5. The discussions and conclusions are given in Section 6.

2. Correntropy-based K-means Given N samples X ¼ fxi gN i ¼ 1 , where each sample is a ddimensional vector, the K-means clustering algorithm aims to find K clusters fμc gKc ¼ 1 by minimizing the objective function K

N

min ∑ ∑ θði; cÞ‖xi  μc ‖22 ;

fμc gKc ¼ 1 c ¼ 1 i ¼ 1

ð1Þ

where θði; cÞ is the membership, which satisfies two conditions, i.e., θði; cÞ A f0; 1g and ∑Kc ¼ 1 θði; cÞ ¼ 1. The objective function of Eq. (1) is sensitive to outliers, because the mean square error (MSE) criterion is used to measure the distance between samples and cluster centers. To tackle this difficulty, the correntropy criterion [33] is adopted instead of the MSE criterion. Thereby, the objective function is reformulated as K

N

min  ∑ ∑ θði; cÞs2 gð‖xi  μc ‖2 Þ;

fμc gKc ¼ 1

ð2Þ

C⋆ ¼

N

min

C;ffμj;c gKc ¼ 1 gj ¼

K

∑ θði; cÞkðj; iÞ‖xi  μj;c ‖22 þ νjCj:

∑ ∑

N

1

The CV model treats segmentation as a global K-means problem. Combining with the contour length term, the segmentation model is C⋆ ¼ min C;fμ

K

K c gc ¼ 1

N

∑ ∑ θði; cÞ‖xi  μc ‖22 þ νjCj;

ð3Þ

c¼1i¼1

where C is the segmentation contour, and ν is a weighting constant of the contour length term jCj. When the input image is distorted by intensity inhomogeneity, pixel values from different clusters overlap, which makes the global method fail. Fig. 1 presents an example to compare the global and local methods. When the image is inhomogeneous, pixel values are hard to separate globally, since they have large overlap. Fortunately, if we consider the pixels in a local region, for example, the pixels contained in the circle, their pixel values can be separated. The local region-based segmentation methods in [19,21] are proposed based on this observation. Li et al. [19] construct the local Kmeans clustering model:1 K

∑

ð5Þ

j ¼ 1 c ¼ 1 iANj

where N is the number of pixels in the whole image.

3.1. The proposed LCK based segmentation model Our LCK based segmentation model is C⋆ ¼

N

C;ffμ

min

N K j;c gc ¼ 1 gj ¼ 1

K

 ∑ ∑

∑ θði; cÞkðj; iÞs2 gð‖xi  μj;c ‖2 Þ þ νjCj:

j ¼ 1 c ¼ 1 iANj

ð6Þ

3. The proposed segmentation method

∑ θði; cÞkðj; iÞ‖xi  μj;c ‖22 ;

fμj;c gKc ¼ 1 c ¼ 1 i A N j

1

is the Gaussian kernel introduced to capture local information, in which dðj; iÞ is the spatial distance between the i-th pixel and the j-th pixel. fμj;c gKc ¼ 1 are the local clusters of the j-th pixel. The segmentation model is constructed by summing all local models together, given by

c¼1i¼1

where gðxÞ ¼ expð  x2 =2s2 Þ is a Gaussian kernel function parameterized with the kernel width s. Due to the Gaussian kernel, the correntropy-based K-means clustering algorithm can adaptively emphasize the samples that are close to their corresponding cluster centers. As a result, the effect of outliers is reduced.

min

where i A N j represents a local pixel around the j-th pixel, and ( ) 1  dðj; iÞ2 exp kðj; iÞ ¼ 2s2d 2π s2d

In [19], this model is named as the local binary fitting (LBF) model.

ð4Þ

Apparently, the main difference between the traditional LBF model (refer to Eq. (5)) and our model (refer to Eq. (6)) is the utilization of pixel-to-cluster distance. This difference is essentially derived from the noise type definition. In LBF model, the noise in each pixel should be Gaussian, while in our model, the noise can be extended to non-Gaussian case. Hence, our LCK based segmentation model is more general.

3.2. Level set formulation In the level set based segmentation method, the segmentation contour C is represented as the zeroth level of a level set function ϕ. We focus on the two-phase segmentation problem.2 In such a case, the cluster number K is set to 2. The membership is defined as

θði; 1Þ ¼ Hðϕi Þ;

θði; 2Þ ¼ 1  Hðϕi Þ;

ð7Þ

where Hð:Þ is the Heaviside function. In practice, the Heaviside function HðÞ is approximated by the ε-Heaviside function   x  1 2 ; ð8Þ H ε ðxÞ ¼ 1 þ arctan 2 π ε and the corresponding derivation is defined by

δε ðxÞ ¼ H ′ε ðxÞ ¼

ε : π ðε2 þ x2 Þ

In our implementation, the parameter

ð9Þ

ε is set to 1.0.

2 By introducing more than one level set function, our model can be extended to the multi-phase case.

L. Wang, C. Pan / Pattern Recognition 47 (2014) 1917–1925

The objective function of our LCK based segmentation model is

ϕ⋆ ¼ ¼

min

Eðϕ; fμj;1 ; μj;2 gN Þ j¼1

min

 ∑ ∑ kðj; iÞs2 ðH ε ðϕi Þgð‖xi  μj;1 ‖2 Þ

ϕ;fμj;1 ;μj;2 gNj ¼ 1

ϕ;fμj;1 ;μ

N

¼

min

ϕ;fμj;1 ;μ

N j;2 gj ¼ 1

∑ ∑ kðj; iÞðH ε ðϕi Þwti ‖xi  μj;1 ‖22

j ¼ 1 iANj

N

N

þ ð1  H ε ðϕi ÞÞwti ‖xi  μj;2 ‖22 Þ þ ν ∑ j∇H ε ðϕj Þj þ κ ∑

N

N j;2 gj ¼ 1

1919

1

j¼12

j¼1

ðj∇ϕj j  1Þ2 :

j ¼ 1 iANj

ð16Þ N

N

1 þ ð1  H ε ðϕi ÞÞgð‖xi  μj;2 ‖2 ÞÞ þ ν ∑ j∇H ε ðϕj Þj þ κ ∑ ðj∇ϕj j 1Þ2 : j¼1 j¼12

ð10Þ In the above equation, the second term is the contour length term, while the third term is the regularization term proposed in [34], which restricts the level set function to a signed distance function.

The level set function and the local clusters are determined from the new objective function of Eq. (16) by the standard gradient descent method. The procedure has the following two sub-steps. t 1 N gj ¼ 1 ,

Step1: For the fixed level set function fϕj fμ

t j;1 ;

μ

t N j;2 gj ¼ 1

the clusters

are calculated by setting the derivatives of Eq. (16)

with respect to μj;1 ; μj;2 equal to zero, t1

4. Model optimization

μtj;1 ¼

The objective function of Eq. (10) is nonlinear, which derives from the nonlinear property of the Gaussian kernel. To solve this problem, we adopt the iteratively re-weighted (IR) algorithm. 4.1. Weight computing in the IR algorithm We first consider the local region around the j-th pixel. In the IR algorithm, the correntropy-based distance is reformulated by the weighted MSE at the t-th iteration, given by s2 gð‖xi  μj;1 ‖2 Þ ¼ wtj;i;1 ‖xi  μj;1 ‖22 ; s2 gð‖xi  μj;2 ‖2 Þ ¼ wtj;i;2 ‖xi  μj;2 ‖22 :

ð11Þ

The two weights wtj;i;1 and wtj;i;2 are determined by3 wtj;i;1 ¼ gð‖xi  μtj;1 1 ‖2 Þ;

wtj;i;2 ¼ gð‖xi  μtj;2 1 ‖2 Þ;

t 1

t1

Þwtj;i;1 þ ð1  H ε ðϕi

ð12Þ

ÞÞwtj;i;2 :

t1

Þwti xi ; t1 Þwti i

μtj;2 ¼

∑i A N j kðj; iÞH ε ðϕ

∑i A N j kðj; iÞð1  H ε ðϕi

ÞÞwti xi : t1 ÞÞwti i

∑i A N j kðj; iÞð1  H ε ðϕ

ð17Þ From Eq. (17), we can see that the local clusters are obtained by calculating the weighted mean. By introducing the weights, the effect of outliers is suppressed. Step2: Keeping the clusters fμtj;1 ; μtj;2 g t N j gj ¼ 1

function fϕ

N j¼1

fixed, the level set

is updated by solving following gradient flow

equation:

ϕ ¼ϕ t j

t1  j

dEðϕ; fμtj;1 ; μtj;2 g

η

dϕj

N j¼1

Þ

;

ð18Þ

η ¼0.1 is the time-step. The gradient of dEðϕ; fμtj;1 ; μtj;2 gNj ¼ 1 Þ=dϕj is

where

where μtj;1 1 and μtj;2 1 are the clusters calculated in the previous time t  1. Eq. (12) shows that each pixel has two weights, namely, wtj;i;1 and wtj;i;2 , which correspond to the two clusters. To obtain a unitary weight for each pixel, we use the following combination:4 wtj;i ¼ H ε ðϕi

∑i A N j kðj; iÞH ε ðϕi

dEðϕ; fμtj;1 ; μtj;2 g dϕj

N j¼1

Þ

¼ H ε ðϕj Þδε ðϕj Þwtj ∑ kðj; iÞð‖xj  μti;1 ‖22  ‖xj  μti;2 ‖22 Þ iANj

!

 νδε ðϕj Þdiv

ϕj ϕj  κ Δϕj  div jϕ j j jϕ j j

!! :

ð19Þ

ð13Þ Algorithm 1. LCK based segmentation.

The final weight of i-th pixel is calculated by summing all local weights together, given by wti ¼ ∑i A N j kðj; iÞwtj;i . Combining with Eqs. (12) and (13), we get t 1

wti ¼ ∑ ðH ε ðϕi iANj

t1

Þgð‖xi  μtj;1 1 ‖2 Þ þ ð1  H ε ðϕi

In our implementation, the threshold

Result: The level set function

Initializing the level set function ϕ ¼ ϕ by Eq. (20);

2:

Initializing the weights fw0j ¼ 1g

ð14Þ

3: 4 5

ð15Þ

6

For t’1 to maxIter do   Calculating the weights fwt gN by Eq: ð14Þ;  j j¼1   N t  Shrinking the weights fwj g by Eq: ð15Þ; j¼1    Computing local clusters fμt ; μt gN by Eq: ð17Þ;  j;1 j;2 j ¼ 1   μt ¼ min fμt ; μt g; μt ¼ max fμt ; μt g;  j;1 j;1 j;2 j;2 j;1 j;2   N t  Updating the level set function fϕj g by Eq: ð18Þ;  j¼1   if ϕt ¼ ¼ ϕt  1 then     break;   end end

7

τ is set to 0.1.

8 9

4.2. Local clusters calculation and level set function updating

10 11

In Eqs. (14) and (15), each pixel is assigned with a weight. Therefore, the objective function at the t-th iteration is

ϕ⋆ ¼

min

ϕ;fμj;1 ;μj;2 gNj ¼ 1

ν and κ.

ϕ.

1

ÞÞgð‖xi  μtj;2 1 ‖2 ÞÞ:

The weight wti is given by ( t wi ; wti Z τ wti ¼ : 0; wti o τ

Data: The input image fxj gN , and parameter j¼1

Eðϕ; fμj;1 ; μj;2 gN Þ j¼1

3 Refer to http://research.microsoft.com/en-us/um/people/zhang/INRIA/Publis/ Tutorial-Estim/node24.html for more information about the weight calculation. 4 The combination is not necessary, we can use two weights for each pixel, which correspond to two clusters.

12

0

N j¼1

;

4.3. Segmentation procedures The procedures of the LCK based segmentation method are summarized in Algorithm 1. The contour is initialized manually.

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L. Wang, C. Pan / Pattern Recognition 47 (2014) 1917–1925

Fig. 2. The segmentation results of our method with different acquisition techniques. The first and third columns are the segmentation results, while the rest columns are their corresponding weight images. The rectangles are initial contours, while the curves are the results. The same as the other figures.

Fig. 3. Our results under different types of images. The first and third columns are the segmentation results, while the rest columns are their corresponding weight images.

The initial level set function 8 > < ρ 0 ϕ ¼ ϕði; t ¼ 0Þ ¼ 0 > :ρ

ϕ0 is obtained by

i A rectangle inner i A rectangle boundary :

5.1. Our results on real images

ð20Þ

i A rectangle outer

As shown in Line 7 of Algorithm 1, the local order regularization proposed in [30] is utilized to improve the stability of our segmentation algorithm. 5. Experimental results Extensive experiments are performed to evaluate our LCK based segmentation method. We compare it with the following approaches, i.e., CV [10],5 LBF [19],6 LIF [21],7 and LBFþ Order [30], quantitatively and qualitatively. 5 6 7

Available at www.mathworks.com/matlabcentral/fileexchange/23445. Available at www.engr.uconn.edu/  cmli/. Available at www4.comp.polyu.edu.hk/  cskhzhang/.

We select five medical images with different acquisition techniques to evaluate our method. The segmentation results are shown in Fig. 2. Although the acquisition techniques are different, our method can still segment the objects successfully. For example, the detailed information is correctly segmented in the MRI brain image and the retinal image, and the weak boundaries are detected in the ultrasound images. Moreover, with the help of the robust correntropy-based measurement, similar background in each image can be suppressed well. Fig. 3 presents our results with different types of images, including medical image, remote sensing image and astronomical image. The results further demonstrate the effectiveness of our algorithm. 5.2. Noise sensitivity comparisons In this experiment, we compare our method with LBF and LBF þOrder on the segmentation of a blood vessel image. The input image is corrupted by the salt and pepper noise with

L. Wang, C. Pan / Pattern Recognition 47 (2014) 1917–1925

LBF

LBF+Order

1921

Our Method

Fig. 4. The comparison of our method with LBF and LBFþ Order. The images in the first column from top to bottom are the original image, the noise distorted image (the noise level is 0.5) and the manually segmentation result. The remaining three columns are the results of LBF, LBFþ Order and our method. For each column, the first row is the segmentation result, while the second and third rows are the two cluster images.

Fig. 5. More comparisons of our method with LBF. The images in first row are the results of LBF, while the image in the second row are our results.

0.5 noise level. That is, half of pixels in this image are corrupted. The input image is preprocessed by performing the median filtering operation. The comparative results are illustrated in Fig. 4. As shown in the first row, despite the input image is greatly corrupted by noise, our LCK based segmentation method can still segment the blood vessel out. However, LBF and LBF þOrder both fail. The second and third rows are two clusters images, i.e.,

background cluster image and foreground cluster image.8 The cluster images provided by our method can preserve edge features better so that our segmented contour can fit the object more 8 All the local clusters fμj;1 gN are reshaped as the background cluster image j¼1 (refer to the images in the second row of Fig. 4). Similarly, all the local clusters fμj;2 gN are reshaped as the foreground cluster image (refer to the images in the j¼1 third row of Fig. 4).

1922

L. Wang, C. Pan / Pattern Recognition 47 (2014) 1917–1925

Noise Level = 0.1

Noise Level = 0.2

Noise Level = 0.3

Noise Level = 0.4

Noise Level = 0.5

Noise Level = 0.6

Noise Level = 0.7

Fig. 6. The comparison of our method with LBFþ Order under different noise levels f0:1; 0:2; …; 0:7g. The images in second row are the LBFþ Order results, while the images in the third row are our results. The images in the first column are the original image and manually segmented results.

Cluster images of LBF+Order

Cluster images of Our Method

Fig. 7. Cluster images comparison. The left two images are cluster images obtained by LBF þOrder. The right two images are obtained by our method.

accurately. Moreover, without the local order regularization, the clusters in the two cluster images are fused together. That is, we can observe large clusters (white dots) and small clusters (black dots) in one cluster image. Due to this, the segmented contour may locate on the smooth regions (refer to the discussions in [30]). Fig. 5 presents more comparisons of our method with LBF. The input images are all distorted by the salt and pepper noise with 0.4 noise level. From this figure, our method provides more accurate segmentation results. In this experiment, we compare our method with LBFþ Order on another blood vessel image under seven noise levels, i.e., f0:1; 0:2; …; 0:7g. Comparative results are shown in Fig. 6. From this figure, we conclude that our segmentation results are more accurate than those of LBF þOrder, especially when the noise level is large. When the noise level reaches to 0.7, our segmented contour cannot fit object edges. Fortunately, as shown in Fig. 7, the two cluster images of our method can preserve the edge features well. On the contrary, it is still impossible for human beings to detect the object from the two cluster images provided by LBFþ Order.

5.3. Segmentation comparisons on real images In this experiment, we compare our method with CV, LIF, LBF and LBFþOrder on a SAR image that is distorted by a strong noise. The comparative results are illustrated in Fig. 8. From this figure, we see that our method segments out the road successfully, yet introduces smaller other clutter background. The success is derived from the utilization of the robust correntropy-based distance measure. Due to this measure, some sporadic pixels, which are not similar with the pixels on the road, are regarded as outliers in the contour evolution. Unfortunately, in LIF, LBF and LBFþ Order, these pixels are considered as foreground at the beginning of the contour evolution, and hence they diffuse to other neighboring pixels during the contour evolution. As a result, the segmentation results of these methods all present undesired background. More comparisons are illustrated in Figs. 9 and 10. From the results on the SAR images in Fig. 9, we see that if the object and the background are similar, both the LBFþ Order and our method can provide good segmentation results. However, if the object and the background are similar, the LBF þOrder fails to segment out

L. Wang, C. Pan / Pattern Recognition 47 (2014) 1917–1925

1923

Fig. 8. The comparison with CV, LIF, LBF and LBFþ Order on a SAR image.

Input Image

LBF+Order

Ours

Weight Image

Fig. 9. Comparisons on two SAR images.

LBF+Order

Ours Fig. 10. The comparison on a ultrasound image.

Weight Image

1924

L. Wang, C. Pan / Pattern Recognition 47 (2014) 1917–1925

Noise Level = 0.01

Noise Level = 0.05

Noise Level = 0.1

Noise Level = 0.2

Noise Level = 0.3

Noise Level = 0.1

Noise Level = 0.2

Noise Level = 0.3

Noise Level = 0.4

Noise Level = 0.5

Fig. 11. An example in the verification image database. The images in the first row are corrupted by Gaussian noise under 5 noise levels, i.e., f0:01; 0:05; 0:1; 0:2; 0:3g, while the images in the second row are corrupted by salt and pepper noise under 5 noise levels, i.e., f0:1; 0:2; 0:3; 0:4; 0:5g.

LIF

LBF

1

1

0.8

0.8

F−score

F−score

CV

0.6 0.4 0.2 0

LBF+Order

Ours

0.6 0.4 0.2

0.01

0.05

0.1

0.2

0.3

Guassian Noise Level

0

0.1

0.2

0.3

0.4

0.5

Salt & Pepper Noise Level

Fig. 12. Quantitatively comparisons by F-score on a verification image database.

the object. The similar observation can be found in a ultrasound image (see Fig. 10). 5.4. Quantitatively comparisons on a verification image database We also evaluate our method on a verification image database, which contains 200 verification images. In practice, the verification images are generated by adding some simulated noise and illumination to character images. Fig. 11 gives an example from this database. To compare the segmentation accuracy quantitatively, we utilize the F-score value as a evaluation criterion, given by Fscore ¼

2  TP ; 2  TP þ FN þ FP

where TP, FP, and FN are the true positive, false positive, and false negative, respectively. A good segmentation result should give a high F-score value. We compare our method with CV, LIF, LBF and LBF þOrder. As shown in Fig. 12, when the testing images are corrupted by Gaussian noise, our F-score values are the same with those of LBFþ Order. When the noise type is salt and pepper, our method is better than others, especially when the noise level is large. This comparative result indicates that our LCK based segmentation method can perform well for images with multiple noise types, especially it performs well with salt and pepper noise.

method is robust to unknown complex noise. Furthermore, as a local method, our method has desirable performance for images with intensity inhomogeneity. Although our LCK based segmentation method is robust to complex noise and intensity inhomogeneity, it also has some limitations. 1. Similar to many traditional region-based level set segmentation methods, the proposed LCK model in our segmentation method is not convex. Hence, our method is a bit sensitive to the initialization, which prevents our method from achieving good results for the automatic segmentation task. 2. In LCK based segmentation, the weights of edge pixels are often decreased faster than those of region pixels during the iterative re-weighting process. Therefore, edge pixels are easier to be classified as outliers than region pixels, which may introduce inaccurate results along image edges. Hence, our method cannot obtain good results for images that contain wispy and clutter targets, such as hair.

In the future, we plan to add shape constraints so that our segmentation method can perform well for special objects as well as handle more complex images. Moreover, as a general method, LCK based clustering can be used for other related applications, such as matting and classification.

6. Discussions and conclusions Conflict of Interest statement In this paper, we propose a robust region-based level set segmentation algorithm based on LCK. Due to LCK, the proposed

None declared.

L. Wang, C. Pan / Pattern Recognition 47 (2014) 1917–1925

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Lingfeng Wang received his B.S. Degree in computer science from Wuhan University, Wuhan, China, in 2007. He is currently an assistant professor Institute of Automation, Chinese Academy of Sciences. His research interests include computer vision and image processing.

Chunhong Pan received his B.S. Degree in automatic control from Tsinghua University, Beijing, China, in 1987, his M.S. Degree from Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, China, in 1990, and his Ph.D. degree in pattern recognition and intelligent system from Institute of Automation, Chinese Academy of Sciences, Beijing, in 2000. He is currently a professor with the National Laboratory of Pattern Recognition of Institute of Automation, Chinese Academy of Sciences. His research interests include computer vision, image processing, computer graphics, and remote sensing.