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Image Segmentation by Combining the Global and Local Properties Article in Expert Systems with Applications · June 2017 DOI: 10.1016/j.eswa.2017.06.008

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Accepted Manuscript

Image Segmentation by Combining the Global and Local Properties Zhenzhou Wang PII: DOI: Reference:

S0957-4174(17)30416-5 10.1016/j.eswa.2017.06.008 ESWA 11378

To appear in:

Expert Systems With Applications

Received date: Revised date: Accepted date:

8 March 2017 5 June 2017 6 June 2017

Please cite this article as: Zhenzhou Wang , Image Segmentation by Combining the Global and Local Properties, Expert Systems With Applications (2017), doi: 10.1016/j.eswa.2017.06.008

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Highlights  We calculate the means of different pixel classes from slope difference distribution.  The image is segmented by clustering the pixels to its nearest mean.  The segmentation is further refined by minimizing the energy of its Gibbs distribution.  The boundary of the refined segmentation is deformed to the nearest detected canny edges.

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Expert Systems With Applications journal homepage: www.elsevier.com

Image Segmentation by Combining the Global and Local Properties Zhenzhou Wang 

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State key Lab of Robotics, Shenyang Institute of Automation, Chinese academy of science (CAS), Shenyang, China

ABSTRACT

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Image segmentation plays a fundamental role in many computer vision applications. It is challenging because of the vast variety of images involved and the diverse segmentation requirements in different applications. As a result, it remains an open problem after so many years of study by researchers all over the world. In this paper, we propose to segment the image by combing its global and local properties. The global properties of the image are characterized by the mean values of different pixel classes and the continuous boundary of the object or region. The local properties are characterized by the interactions of neighboring pixels and the image edge. The proposed approach consists of four basic parts corresponding to the global or local property of the image respectively: (1), the slope difference distribution that is used to compute the global mean values of different pixel classes; (2), Energy minimization to remove inhomogeneity based on Gibbs distribution that complies with local interactions of neighboring pixels; (3), the Canny operator that is used to detect the local edge of the object or the region; (4), the polynomial spline that is used to smooth the boundary of the object or the region. These four basic parts are applied one by one and each of them is indispensable for the achieved high accuracy. A large variety of images are used to validate the proposed approach and the results are favorable.

2017 Elsevier Ltd. All rights reserved.

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Keywords: Segmentation; Histogram; Gibbs distribution; Slope difference distribution

 Corresponding author. Tel.: +86-24-2397-0612; Fax. +86-024-23970021; e-mail: [email protected]

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Image segmentation algorithms are usually only effective for some specific types of images while they might make great mistakes for other types of images due to four major problems. (1), the histogram distributions of different pixel classes overlap with each other, which makes the estimation of the global parameters inaccurate. (2), the noise that modifies the histogram distribution might cause great errors during global parameter estimation. (3), the local properties of different regions in the image are usually not homogeneous at the same spatial scale, which makes it difficult for modeling-based segmentation methods to estimate the local parameters with satisfactory accuracy. (4), the overlap of different grayscale distributions makes the boundaries of the labeled regions inaccurate. The first two problems persist in the famous expectation maximization (EM) algorithm (Dempster et al., 1977; Besag, 1986) that has been widely used to estimate the means and variances of different pixel classes during image segmentation although its effectiveness has been widely accepted (Carson et al., 2002; Won and Gray, 2004). The other popular parameter estimation algorithm, k-means clustering method (Jain, 2010) could not effectively deal with the first problem and the second problem either. The estimated global parameters by EM or Kmeans are frequently inaccurate in image segmentation applications. For the popular thresholding methods (OsunaEncisa et al., 2013), their accuracies are also affected by the first two problems. Some researchers have tried to solve the second problem by estimating the noise separately and apply the denoising technique to suppress the estimated noise (Sagar et al., 2014, Manjon et al., 2015, Chan et al., 2006), which achieved significant improvement in segmentation accuracy. However, it is also very difficult to estimate the parameters of the noise model accurately. The overlap of the gray-scale distributions of different pixel classes, the different levels of noise magnitudes and the different forms of noise distributions make it difficult to estimate the parameters with adequate accuracy in most cases. To address the third problem, the image had been assumed as a Markov Random Field (MRF) that is subject to the Gibbs distribution (Derin and Elliott, 1987; Caillol et al., 1993; Wang and Zhang, 2012). Based on this assumption, a lot of efforts have been put into the segmentation research work and many MRFbased segmentation methods have been proposed. For instance, some researchers proposed to cluster the texture features with a mixture of Gaussian models for better performance (Yang et al., 2008). Some researchers combined the Maximum a posteriori (MAP), maximum likelihood (ML) and graph cut to achieve the satisfactory segmentation results (Chen et al., 2010). One merit of the MRF-based method is that it could distinguish different textured regions, especially for the textures that could be modeled and defined with parameters. The major drawback of MRF-based methods is that they could not estimate the parameters of the textures with adequate accuracy in most cases, especially for the natural images. On the other hand, its segmentation accuracy is determined by the accuracy of the parameter estimation. Consequently, the accuracy of MRF-based methods is only good for a specific class of images whose texture parameters could be estimated robustly. Although the parameter of the Gibbs distribution might not be estimated accurately in most cases, it characterizes the interaction of neighboring pixels well and thus could be utilized in the segmentation process. Instead of estimating its parameters, we use the iterative gradient descent method to minimize the energy of the Gibbs distribution until it converges to a steady state. For the fourth problem, we did not find any literature that is specially devoted to addressing it though we have found the segmentation artifacts caused by it in some popular methods (Carson et al., 2002; Felzenszwalb and Huttenlocher, 2004). The methods based on variational formulations (Chan and Vese, 2001; Li et al., 2010; Lankton and Tannenbaum, 2008; Arbelaez et al., 2011) are not affected by the fourth problem. However,

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none of these methods is good at segmenting a variety of images. They usually perform well in a class of images while performs badly for another class of images. Based on all these facts, it is necessary to explore new ways or directions for image segmentation to achieve better overall performance. In this paper, we propose a new image segmentation approach that could deal with these four problems effectively. (1), to reduce the interactions between different distributions and to estimate the global parameters of the image robustly, we estimate the means of different pixel classes by the global peaks of the histogram distribution. These peaks are isolated from each other and are not affected by the overlap of the grayscale distributions of different pixel classes. (2) to reduce the noise, we smooth the histogram distribution with a Fourier transformation based filter that is proven to be more effective than other filters. (3), to make use of the local property of the image, we propose an iterative gradient descent method to minimize the global Gibbs energy until it converges. (4), to improve the accuracy of the extracted boundaries from the labeled regions, we deform them to the nearest detected Canny edges and then use the polynomial interpolation to smooth them. To detect the global peaks from the smoothed histogram distribution robustly, we utilize the slope difference distribution that is defined as the variation rate of the histogram distribution (Wang, 2016). The peaks and valleys of the histogram distribution usually have much greater variation rates compared to other parts of the histogram. In addition, the global peaks of the histogram distribution correspond to the cluster centers of the pixel classes when they are Gaussian distributed, which is true in most image segmentation cases. The slope difference distribution is computed by calculating the difference of the slope values on the left side and the right side of each point on the smoothed histogram. The slope values are computed by fitting two straight lines on both sides of the point on the smoothed histogram with N points. The proper selection of N could suppress smaller local peaks while retain larger global peaks. The means of different pixel classes correspond to the global peaks of the slope difference distribution distinctly and are resistant to the problem caused by the distribution overlap of different pixel classes. This paper is organized as follows. In section 2, we analyzed the problems of image segmentation and described the proposed approach. In section 3, both synthesized images and natural images are segmented to validate the proposed approach. Discussions are given in section 4. Section 5 concludes the paper.

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1. Introduction

2. The proposed approach The proposed approach deals with these four described challenging problems one by one. Firstly, the means of different pixel classes are computed from the global peaks of the histogram distribution. The advantages of computing the mean values of the pixel classes based on the global peaks of the histogram distribution include: (1), If we calculate the cluster centers of the pixel classes from the global peaks in the histogram distribution, the overlapping parts of different pixel classes at the valley parts of the histogram will not affect the calculation accuracy. (2), Since the peaks of the histogram distribution are isolated from each other, they will not affect each other’s parameter calculation accuracy either. As the estimation of the probability distribution of a continuous variable, the histogram distribution of the image plots the number of pixels for each gray-scale value. It was utilized by different researchers in image quality evaluation, pixel clustering and image segmentation. However, its functionality is usually limited by the complexity of the image and the noise that modifies the histogram. As a result, it is usually difficult to compute the global peaks from the histogram distribution directly. In this paper, we compute the slope difference distribution (Wang, 2016) from the histogram distribution first and then calculate the global peaks of the histogram distribution from the peaks of the slope difference distribution. The calculation of the slope difference distribution is summarized as follows.

P  x  i 

Ni ; i  G1 ,..., G2 Nj

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j  argmax  N k 

(2)

k G1 ,G2 

where N i denotes the frequency of the gray-level i and N j denotes the maximum frequency which occurs at j in the interval [G1 , G2 ] . G1 is the minimum gray-level of the image and

G2 is the maximum gray-level of the image. For any point, i on the histogram distribution, there are two slopes, one on the left and the other on the right. We compute the slope by fitting a straight line with N adjacent points on each side of the point, i . The linear model of the straight line is formulated as: (3) yi  axi  b

 a, b

T

  BT B  BT Y

 x1  x B  2    xN

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The number N of the fitting points in the line model is critical for magnifying the global peaks while suppressing the local peaks. Thus, its selection is very important and needs some calibration process to determine its value for a specific class of images. Fortunately, there is a wide range of N values that could meet the requirement and its default value is chosen as 15 for the histogram in the interval of [0, 255]. The positions where the global peaks occur correspond to the mean values of the pixel classes in the image. To compute the peaks’ positions from the slope difference distribution, we set the derivative of the slope difference distribution s  x  to zero and

Synthesized image with noise level     0,   20  ; (c) Finding the global peaks by slope difference distribution computed for the first synthezied image; (d) Finding the global peaks by slope difference distribution computed for the second synthezied image.

From Fig. 1 (d), we see the second problem that the noise modifies the histogram distribution and causes great errors during calculation of the pixel mean values. To eliminate or reduce the bad impact of noise, we propose the following low- filtering method in the frequency domain. The histogram distribution, P  x  is transformed into the frequency domain by the Discrete

F k  

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 0  50, 0  10; 1  100,1  10; 2  150, 2  10 .

G2

P  x e

i

2 kx T

; k  G1 ,..., G2

(9)

x  G1

We obtain the candidate mean values of different pixel classes by solving the above equations. We define the peaks whose magnitudes are greater than a threshold (0.05 in this study) as the valid peaks. The candidate means that correspond to the valid peaks are sorted. The number of pixel classes is determined by the total number of the sorted means or manual input. When the number of pixel classes is determined manually, the means are selected one by one based on their magnitudes. The mean value that corresponds to the peak with the greatest magnitude is selected as the first mean. To demonstrate the process of finding the global peaks by slope difference distribution, we synthesized two images with two objects and one background in Fig. 1 (a) and (b) respectively. The synthesized image in Fig. 1 (a) is with the pixel mean value of the background equal 50, the pixel mean value of the first object equal 100 and the pixel mean value of the second object equal 150. For the first synthesized image, we add the Gaussian noise with the magnitude     0,   10  and the image model is

(c) (d) Fig. 1. Demonstration of finding the global peaks by the slope difference distribution. (a) Synthesized image with noise level     0,   10  ; (b)

Fourier Transformation (DFT):

select the solutions corresponding to the peak parts.

ds  0; s( x)  0 dx

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(6)  y1 , y2 ,, yN  Two slopes at point i , a1  i  and a2  i  are obtained from Eq. (4). The slope difference, s  i  , at point i is computed: s  i   a2  i   a1 i  ; i  G1  N ,..., G2  N (7)

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computed by Eq. (1) is drawn in green, the slope difference distribution computed by Eq. (7) is drawn in blue. The blue crosses corresponding to the global peaks are computed by Eq. (8). In Fig. 1 (c), the slope difference distribution has three peaks that correspond to the mean values of the background, the dark object and the bright object respectively. Because of the relatively smaller noise magnitude, the computed slope difference distribution is smooth and the global peaks could be detected by Eq. (8). On the contrary, the noise magnitude in Fig. 1 (b) is relatively larger. Consequently, the computed slope difference distribution is not smooth enough for Eq. (8) to detect the global peaks correctly.

(4)

1  1   1

The

process of finding the global peaks by the slope difference distribution for each synthesized image is demonstrated in Fig. 1 (c) and (d) respectively. The histogram distribution P  x 

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The histogram distribution

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For

the second synthesized image, we add the Gaussian noise with the magnitude     0,   20  and the image model is

where

T  G2  G1  1

(10)

In the frequency domain, the low frequency parts of F  k  are kept while the high frequency parts of it are eliminated by the following equation.  F  k  ; k  G1 , G1  1,..., G1  B  1  (11) F   k    F  k  ; k  G2  B  1,..., G2  1, G2  0; k  G1  B,..., G2  B  where B denotes the bandwidth of the low filter and we set its default value as 10 based on testing a great variety of images. After transforming the histogram from the frequency domain back into the time domain by the following equation, we get the smoothed histogram, P  x  .

P  x  

2 xk i 1 G2 F   k  e T ; x  G1 ,, G2  T k G1

(12)

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To demonstrate the effectiveness of the proposed low- filtering method, we show the process of finding global peaks by the slope difference distribution computed from the smoothed histogram distribution for the synthesized image (Fig. 1 (b)) in Fig. 2. As can be seen, the proposed filter could remove the noise from the histogram distribution effectively and the global peaks could be detected correctly after the slope difference distribution was computed from the smoothed histogram distribution. The reason we use the Fourier transform based low filter instead of other filters is that it is capable of preserving the shape of the low frequency part well while removing the high frequency parts effectively. After the mean values of all the pixel classes, l ; l  1,.., L are computed, the image is labeled by the following equation. Ilabel (u, v)  l; u  1,..,U ; v  1,..,V (13)

that is computed as:

l  arg min I (u, v)  l

(a) (b) Fig. 5. Comparison of boundary with detected canny edges. (a) Extracted boundary overlaying on the detected canny edges; (b) Final smooth boundary overlaying on the detected canny edges.

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(a) (b) Fig. 3. Demonstration of labelling the image. (a) The labelled image by Eqs. (13-14); (b) The labelled image after Gibbs energy minimization.

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We computed the labeled result of the synthesized image (Fig. 1 (b)) by Eqs. (13-14) and showed it in Fig. 3 (a). It is seen that the labeled regions are inhomogeneous, which is caused by the third challenging problem that the local properties of different regions in the image are usually not homogeneous at the same spatial scale. This local property could be formulated by the Gibbs distribution that characterizes the image as a Markov random field X = {Xs} with a t distribution (Derin and Elliott, 1987; Caillol et al., 1993; Wang and Zhang, 2012).



1  V ( x) e  cC c V ( x)  cC c e 

xL

(15)

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P( X  x) 

where Vc ( x) is the potential function associated with clique c and it is defined as follows:

  ; all values in c are equal Vc ( x)   (16)   ; else

where  is a constant and its default value is 1. To eliminate the inhomogeneity in the labelled image, we utilize the Gibbs distribution to minimize its total energy over the whole image.

I o  arg min

L

U

V

  E( I

Ilabel ( u , v )[1, L ] l 1 v 1 v 1

label

(u, v))

E ( Ilabel (u, v))   cC Vc ( x)

(a) (b) Fig. 4. Demonstration of boundary delineation. (a) Extracted boundary from the labelled image with Gibbs energy minimization; (b) Delineated boundary after deforming the bounday in (a) to the nearest Canny edges.

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where (u, v) denotes the position of the pixel in the image and U V denotes the resolution of the image. L denotes the total number of pixel classes in the image and l denotes the pixel class

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Fig. 2. Demonstration of the effectiveness of the proposed filter and the correctness of finding the global peaks by the computed slope difference distribution.

eliminated successfully. For the practical implementation of the Gibbs energy minimization, please refer Wang and Zhang, 2012. After the image is labeled and its inside inhomogeneity is eliminated, the boundaries of the labeled objects are extracted as shown in Fig. 4 (a). It is seen that the Gibbs energy minimization causes the boundary to be noisy and the extracted boundary does not match the object’s true boundary well. In addition, there are two different boundaries extracted for the brightest object, which is caused by the fourth challenging problem that the boundary of the labeled region is affected greatly by the overlap the grayscale distributions of different pixel classes. We define the problem that multiple boundaries are extracted from the segmentation result for the same object as the pixel distribution overlapping problem. This problem could not be avoided for the labelling method based on the Euclidean distance (Eqs. (13-14)) and it frequently appears in other literatures (Carson et al., 2002; Felzenszwalb and Huttenlocher, 2004).

(17) (18)

where I o is the labeled image with Gibbs energy minimization and Fig. 3 (b) shows the labelled image with Gibbs energy minimization for the synthesized image in Fig. 1 (b). As can be seen, the inhomogeneity inside each region has been

To refine the extracted boundary B , we propose the following boundary deformation method. Step 1: the edge of the image is detected with the canny operator (Canny, 1986). Step 2: the extracted boundaries are deformed to the nearest detected edges by minimizing the following energy function. R

 arg min B(r )  E(q) r 1 q[1,2...,Q ]

(19)

where B denotes the boundary extracted from the segmented object by Eqs (1)-(18) . R and r denote the total number of points and the index of the points on it respectively. E denotes the detected canny edges. Q and q denote the total number of points and the index of the points on it respectively. The total number of points R on the canny edges are usually much greater than the total number of the points Q on the extracted boundary. There exist R points on the extracted canny edges that make Eq. (19) minimum and they form the deformed boundary that is denoted as E . As can be seen, the points on the deformed boundary are all from the detected canny edges. The final smooth boundary Bs is computed by minimizing the energy between the deformed boundary E and the fitted polynomial spline. 2

2

E ( Bs )  (1   )  Bs ( j )  E ( j ) dj   

d 2 Bs (t ) dt (20) dt 2

where  is the smoothing factor and its default value is 0.5. Fig. 4 (b) shows the final smooth boundary obtained from the extracted boundary in Fig. 4 (a) by Eqs. (19-20). As can be seen, the final smooth boundary match the object’s true boundary well and the accuracy of the extracted boundary is improved significantly. To show the matching degree of the detected Canny edges with the extracted boundaries from the labelled objects and the final smooth boundaries respectively, we show the overlay of them in Fig. 5 (a) and Fig. 5 (b) respectively. In (a), most of the extracted boundaries from the labelled objects do not match the

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detected Canny edges while most of the final smooth boundaries match the detected Canny edges well in (b). 3. Experimental results

acceptable. This demonstration indicates that there is a wide range of the bandwidth for the Fourier transform based low- filter to achieve an acceptable performance (at least the range [5 40] meets the requirement).

3.1. Performance analysis The proposed method depends on the fitting number N, the Fourier transform based low- filter and the Canny edge to achieve the optimal segmentation performance. In this section, we will analyze the performance of the proposed method in detail with demonstrations. Firstly, we demonstrate how the fitting number N affects the segmentation performance in Fig. 6. We change the magnitude of the Gaussian noise of the synthesized image to     0,   15 and segment it by the proposed

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method with different values of N. The fitting number N is chosen as 5, 15, 25 and 50 respectively. Their corresponding slope difference distributions are plotted in Fig. 6 (a), (b), (c) and (d) respectively. The corresponding segmentation results are shown in Fig. (e), (f), (g) and (h) respectively. As can be seen, the number of the peaks and valleys of the slope difference distributions decrease with the increase of the value of N. The automatic computed clustering centers are correct when N equals 5, 15 and 25. However, when N equals 45, the computed clustering center is incorrect. This demonstration indicates that there is a wide range of N that could meet the requirement to make the proposed method to achieve an acceptable performance (at least the range [5 25] meets the requirement).

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(e) (f) (g) (h) Fig. 6. Demonstration of how the fitting number N affects the segmentation performance with the systhesize image (its noise level equals     0,   15 ). (a) clustering results of the slope difference distribution when N equals 5; (b) clustering results of the slope difference distribution when N equals 15; (c) clustering results of the slope difference distribution when N equals 25; (d) clustering results of the slope difference distribution when N equals 45; (e) the segmentation result when N equals 5; (f) the segmentation result when N equals 15; (g) the segmentation result when N equals 25; (h) the segmentation result when N equals 45.

Secondly, we demonstrate how the Fourier transform based low- filter affects the segmentation performance. From Eq. (11), it is seen that the bandwidth of the filter determines the output of the filter. Hence, we use different bandwidth values to compare the segmentation performance. We select the bandwidth as 5, 20, 40 and 50 respectively. The clustering results by slope difference distribution and the final segmentation results are shown in Fig. 7 (a)-(h) respectively. The final segmentation results are acceptable when the bandwidth equals 5, 20 and 40. However, when the bandwidth equals 50, the segmentation is not

(e) (f) (g) (h) Fig. 7. Demonstration of how the bandwidth of the Fourier transform based low- filter affects the segmentation performance with the systhesize image (its noise level equals     0,   15 ). (a) clustering results of the slope difference distribution when the bandwidth equals 5; (b) clustering results of the slope difference distribution when the bandwidth equals 20; (c) clustering results of the slope difference distribution when the bandwidth equals 40; (d) clustering results of the slope difference distribution when the bandwidth equals 50; (e) the segmentation result when the bandwidth equals 5; (f) the segmentation result when the bandwidth equals 20; (g) the segmentation result when the bandwidth equals 40; (h) the segmentation result when the bandwidth equals 50.

Thirdly, we demonstrate how the Gaussian noise level affects the segmentation performance in Fig. 8. The Gaussian noise level will affect the detected edges by Canny operator and the clustering accuracy of slope difference distribution. We select the noise level of the synthesized image as 20, 30, 40 and 50 respectively. The detected Canny edges and the final segmentation results for these four synthesized images are shown in Fig. 8 (a)-(h) respectively. As can be seen, the quality of the detected Canny edges decreases with the increase of the noise level, so does the segmentation performance. However, the increase of the noise level has more adverse effect on the clustering accuracy of slope difference distribution than on the detected Canny edges. Hence, we conclude that the Canny detection could withstand the Gaussian noise level as long as the slope difference distribution could withstand. In addition, noise will cause burrs on the detected Canny edges, which will decrease the segmentation performance.

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(e) (f) (g) (h) Fig. 8. Demonstration of how the Gaussian noise level affects the segmentation performance with the systhesize image. (a) canny edges of the

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noise magnitude     0,   30  ; (d) final segmentation result of the synthesized image with Gaussian noise magnitude     0,   30  ; (e) canny edges of the synthesized image with Gaussian noise magnitude     0,   40  ; (f) final segmentation result of the synthesized image with Gaussian noise magnitude     0,   40  ; (g) canny edges of the synthesized image with Gaussian noise magnitude     0,   50  ; (h) final segmentation result of the synthesized image with Gaussian noise magnitude     0,   50  .

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(e) (f) (g) (h) Fig. 9. Demonstration of how the Salt and Pepper noise level affects the segmentation performance with the systhesize image. (a) canny edges of the synthesized image with Salt and Pepper noise density 0.8; (b) final segmentation result of the synthesized image with Salt and Pepper noise density 0.8; (c) canny edges of the synthesized image with Salt and Pepper noise density 1.6; (d) final segmentation result of the synthesized image with Salt and Pepper noise density 1.6; (e) canny edges of the synthesized image with Salt and Pepper noise density 2.4; (f) final segmentation result of the synthesized image with Salt and Pepper noise density 2.4; (g) canny edges of the synthesized image with Salt and Pepper noise density 3.2; (h) final segmentation result of the synthesized image with Salt and Pepper noise density 3.2.

To prove the DFT low filter is good at preserving the shape of the low frequency part well, we design the following experiments. We compare it with other well-known filters quantitatively in maintaining the means of the pixel classes. We vary the magnitude of the Gaussian noise from     0,   10 

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Table 1. Comparison of the accuracy of computing the clustering centers by the proposed approach with different filters Filters\Ground Truths 50 100 150 butter-worth filter

60

108

157

Chebyshev Type I filter

59

107

156

Chebyshev Type II filter

55

104

153

Elliptic filter

56

105

153

FIR low filter

51

99

149

moving average filter

51

99

149

DFT filter

50

100

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3.2. Results of segmenting synthesized images

To control the image model precisely, we synthesized several images to demonstrate the advantage of the proposed approach over state of the art methods. We compared the performance of the proposed approach with eleven methods with available MATLAB codes: (1), Kernel Graph Cuts (Salah et al., 2011); (2), Normalized Graph Cuts (Shi and Malik, 2000); (3) Graph (Felzenszwalb and Huttenlocher, 2004); (4), EM (Dempster et al., 1977); (5), K-means (Jain, 2010); (6), Active contours without edge (Chan and Vese, 2001); (7), GM (Chan et al., 2006); (8), Distance regularized level set (Li et al., 2010); (9), Region based active contour (Lankton and Tannenbaum, 2008); (10), GMM (Nguyen et al., 2013); (11) FFCM (Nguyen and Wu, 2013). We set the iteration of active contour and level set methods as 2000 and the number of N in fitting straight lines during computing the slope difference distribution as 15 in all the experiments conducted in this research work. We select the same number of the pixel classes for the proposed approach, K-means method and EM method in segmenting the tested images. The parameters of other state of the art methods are kept as their default values.

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We also demonstrate how the Salt and Pepper noise affects the segmentation performance in Fig. 9. We select the Salt and Pepper noise density of the synthesized image as 0.8, 1.6, 2.4 and 3.2 respectively. The detected Canny edges and the final segmentation results for these four synthesized images are shown in Fig. 9 (a)-(h) respectively. As can be seen, the quality of the detected Canny edges decreases with the increase of the noise level, so does the segmentation performance. However, the increase of the noise level has more adverse effect on the detected Canny edges than on the clustering accuracy of slope difference distribution. Hence, we conclude that the canny detection could withstand less Salt and Pepper noise level than the slope difference distribution could withstand.

approach depends on the shape of the filtered histogram, thus we say that the shape of histogram is kept well by the DFT low filter.

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synthesized image with Gaussian noise magnitude     0,   20  ; (b) final segmentation result of the synthesized image with Gaussian noise magnitude     0,   20  ; (c) canny edges of the synthesized image with Gaussian

to     0,   20  and get 10 synthesized images. We then use different low filters to filter the histogram for the proposed approach and compute the clustering centers for these 10 synthesized images. For any low filter, the computed clustering centers almost keep the same for these 10 synthesized images. We use their averages to compare the performances of different low filters. The comparison is shown in Table 1. The compared filters include Infinite Impulse Response (IIR) butter-worth low filter, IIR Chebyshev Type I low filter, IIR Chebyshev Type II low filter, IIR Elliptic low filter, FIR low filter and the time domain moving average filter. Only when the histogram is filtered by the designed DFT low filter, the proposed approach could find the clustering centers accurately, which indicates that the centers of the pixel classes are kept well by the DFT low filter. Because the calculation of the clustering centers by the proposed

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(g) (h) (i) (j) (k) (l) Fig.10. Results of segmenting synthesized noisy images; (a) Proposed; (b) Kernel Graph Cuts (Salah et al., 2011); (c) Normalized Graph Cuts (Shi and Malik, 2000); (d) Graph (Felzenszwalb and Huttenlocher, 2004); (e) EM (Dempster et al., 1977); (f) K-means (Jain, 2010); (g) Active contours without edge (Chan and Vese, 2001); (h) GM (Chan et al., 2006); (i) Distance regularized level set (Li et al., 2010); (j) Region based active contour (Lankton and Tannenbaum, 2008); (k) GMM (Nguyen et al., 2013); (l) FFCM (Nguyen and Wu, 2013).

We synthesize an image with three ellipse objects on the background in Fig. 10 and their means and variances are {200, 30}, {150, 30}, {100, 30} and {50, 30} respectively. As can be seen, the proposed approach, Normalized graph cuts and k-means perform better than other methods for this type of images. Distance regularized level set and region based active contour are affected by the noises greatly. Compared to them, GMM, FFCM, active contour without edge and GM are more accurate and efficient. However, they could only separate one ellipse successfully. Kernel graph cuts and EM fail to separate the smallest ellipse from the background. Graph segmented the image into too much small regions while the boundaries of the segmented ellipses are not accurate.

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Proposed; (b) Kernel Graph Cuts (Salah et al., 2011); (c) Normalized Graph Cuts (Shi and Malik, 2000); (d) Graph (Felzenszwalb and Huttenlocher, 2004); (e) EM (Dempster et al., 1977); (f) K-means (Jain, 2010); (g) Active contours without edge (Chan and Vese, 2001); (h) GM (Chan et al., 2006); (i) Distance regularized level set (Li et al., 2010); (j) Region based active contour (Lankton and Tannenbaum, 2008); (k) GMM (Nguyen et al., 2013); (l) FFCM (Nguyen and Wu, 2013)..

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3.3. Results of segmenting images from Berkeley dataset

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(g) (h) (i) (j) (k) (l) Fig.12. Results of segmenting another synthesized noisy images; (a) Proposed; (b) Kernel Graph Cuts (Salah et al., 2011); (c) Normalized Graph Cuts (Shi and Malik, 2000); (d) Graph (Felzenszwalb and Huttenlocher, 2004); (e) EM (Dempster et al., 1977); (f) K-means (Jain, 2010); (g) Active contours without edge (Chan and Vese, 2001); (h) GM (Chan et al., 2006); (i) Distance regularized level set (Li et al., 2010); (j) Region based active contour (Lankton and Tannenbaum, 2008); (k) GMM (Nguyen et al., 2013); (l) FFCM (Nguyen and Wu, 2013)..

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We synthesize a more complex image with four rectangular objects on the background in Fig. 12 and their means and variances are {230, 30}, {180, 30}, {140, 30}, {90, 30} and {40, 30} respectively. As can be seen, only the proposed approach could segment the rectangles successfully while all the other methods perform badly, but differently. In addition, some state of the art methods performed relatively better than other state of the art methods. For instance, FFCM performs the second best in this specific case. We blur the synthesized image in Fig. 12 by minimizing the Gibbs distribution of the image again to get next fuzzy image. The segmentation results are shown in Fig. 13 (a)-(j). At this time, both the proposed approach and the EM method could segment the four rectangles successfully while the proposed approach has better accuracy than the EM method. All the other tested state of the art methods failed to give acceptable segmentation results in this specific example. From all the synthesized images, we could see that the proposed approach is more robust than state of the art methods.

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(g) (h) (i) (j) (k) (l) Fig.14. Results of segmenting a typical image from Berkeley dataset; (a) Proposed; (b) Kernel Graph Cuts (Salah et al., 2011); (c) Normalized Graph Cuts (Shi and Malik, 2000); (d) Graph (Felzenszwalb and Huttenlocher, 2004); (e) EM (Dempster et al., 1977); (f) K-means (Jain, 2010); (g) Active contours without edge (Chan and Vese, 2001); (h) GM (Chan et al., 2006); (i) Distance regularized level set (Li et al., 2010); (j) Region based active contour (Lankton and Tannenbaum, 2008); (k) GMM (Nguyen et al., 2013); (l) FFCM (Nguyen and Wu, 2013)..

Fig. 14 (a)-(j) showed the results of segmenting the plane from the sky background by different methods. As can be seen, the kernel graph cuts and normalized graph cuts methods tend to segment the image into some regions unexpectedly. The Graph, EM and K-means methods yield too many regions. The active contour without edge, GM, GMM, FFCM, Distance regularized level set and region based active contour methods failed to delineate the boundaries of the plane accurately. The result of the proposed approach appears to be better. We compare the segmentation results on another 3 images in Figs. 15-17. The visual results verified the superiority of the proposed approach over state of the art methods.

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(g) (h) (i) (j) (k) (l) Fig.15. Results of segmenting a typical image from Berkeley dataset; (a) Proposed; (b) Kernel Graph Cuts (Salah et al., 2011); (c) Normalized Graph Cuts (Shi and Malik, 2000); (d) Graph (Felzenszwalb and Huttenlocher, 2004); (e) EM (Dempster et al., 1977); (f) K-means (Jain, 2010); (g) Active contours without edge (Chan and Vese, 2001); (h) GM (Chan et al., 2006); (i) Distance regularized level set (Li et al., 2010); (j) Region based active contour (Lankton and Tannenbaum, 2008); (k) GMM (Nguyen et al., 2013); (l) FFCM (Nguyen and Wu, 2013)..

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We blur the synthesized image in Fig. 10 by minimizing the Gibbs distribution of the image to get the fuzzy image. The segmentation results are shown in Fig. 11 (a)-(j). At this time, kmeans failed to segment the image correctly while the proposed approach and normalized cuts still perform well. All the other tested methods perform unsatisfactorily.

In this section, we show the results of segmenting some typical images from the Berkeley dataset (Martin, D. and Fowlkes, C.) to the superiority of the proposed approach over state of the art methods. The color image is converted to the grayscale image by selecting the first RGB image channel directly and then the proposed approach and state of the art methods are applied on the grayscale image one by one.

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(g) (h) (i) (j) (k) (l) Fig.11. Results of segmenting synthesized fuzzy images; (a) Proposed; (b) Kernel Graph Cuts (Salah et al., 2011); (c) Normalized Graph Cuts (Shi and Malik, 2000); (d) Graph (Felzenszwalb and Huttenlocher, 2004); (e) EM (Dempster et al., 1977); (f) K-means (Jain, 2010); (g) Active contours without edge (Chan and Vese, 2001); (h) GM (Chan et al., 2006); (i) Distance regularized level set (Li et al., 2010); (j) Region based active contour (Lankton and Tannenbaum, 2008); (k) GMM (Nguyen et al., 2013); (l) FFCM (Nguyen and Wu, 2013).

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(g) (h) (i) (j) (k) (l) Fig.17. Results of segmenting a typical image from Berkeley dataset; (a) Proposed; (b) Kernel Graph Cuts (Salah et al., 2011); (c) Normalized Graph Cuts (Shi and Malik, 2000); (d) Graph (Felzenszwalb and Huttenlocher, 2004); (e) EM (Dempster et al., 1977); (f) K-means (Jain, 2010); (g) Active contours without edge (Chan and Vese, 2001); (h) GM (Chan et al., 2006); (i) Distance regularized level set (Li et al., 2010); (j) Region based active contour (Lankton and Tannenbaum, 2008); (k) GMM (Nguyen et al., 2013); (l) FFCM (Nguyen and Wu, 2013)..

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(g) (h) (i) (j) (k) (l) Fig.16. Results of segmenting a typical image from Berkeley dataset; (a) Proposed; (b) Kernel Graph Cuts (Salah et al., 2011); (c) Normalized Graph Cuts (Shi and Malik, 2000); (d) Graph (Felzenszwalb and Huttenlocher, 2004); (e) EM (Dempster et al., 1977); (f) K-means (Jain, 2010); (g) Active contours without edge (Chan and Vese, 2001); (h) GM (Chan et al., 2006); (i) Distance regularized level set (Li et al., 2010); (j) Region based active contour (Lankton and Tannenbaum, 2008); (k) GMM (Nguyen et al., 2013); (l) FFCM (Nguyen and Wu, 2013)..

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called VoI and the lower the better. We use the quantitative results from the literatures directly instead of applying these state-of-the-art methods on the Berkeley datasets to avoid abusing these methods. The comparison is also favorable and the proposed approach achieves better accuracy in both measures. For those methods that we could not find the quantitative results from the literatures, we design the following experiments to acquire the quantitative comparisons. Firstly, we vary the magnitude of the Gaussian noise of the synthesized image in Fig. 1 from 1 to 100. Then we de-noise the synthesized noisy images by minimizing the Gibbs energy and generate the fuzzy images. We segment the two objects from the background by different methods and compare the computed F-measures. The computed F-measures for the 100 synthesized images by different methods are plotted in Fig. 20. The average F-measures computed by different methods are shown in Table. 4. As can be seen, the proposed method achieves significantly better accuracy. We synthesize another 100 images with only one object and segment the object from the background by different methods. The computed F-measures are compared in Fig. 21 and the average Fmeasures are compared in Table. 5. As can be seen, the proposed method also achieves better accuracy. We show the ground truth boundaries for the first dataset and for the second dataset in Fig. 22 (a) and (b) respectively for better understanding of the designed experiments. The ground truth boundaries are computed when the noise level is zero and they are overlaying on the synthesized fuzzy image with the magnitude of the Gaussian noise equal 10.

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In this section, we evaluate the proposed approach quantitatively. In this study, we first compare the accuracy of the proposed approach with that of the EM and K-means methods in finding the pixel mean values. We synthesize an image dataset with 100 images in total. In the synthesized dataset, one hundred images were generated with the global parameter 𝜃y = 0  50,V ; 1  100, V ; 2  150, V; 3  200, V  and V varies

(a) (b) Fig.18. Demonstration of the synthesized images in the dataset (a) The synthesized noisy image; (b) The fuzzy image obtained by minimizing the Gibbs energy of the synthesized image in (a).

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from 10 to 110. In the synthesized image, there are four pixel classes with mean values equal to 50, 100, 150 and 200 respectively and their variances are denoted by V . Fig. 18 (a) shows the first synthesized image with V equal to 10. Before computing the pixel mean values by the compared methods, we de-noise the synthesized noisy images by minimizing the Gibbs energy first and produce the corresponding fuzzy images. Fig. 18 (b) shows a produced fuzzy image after de-noising the synthesized image in (a). Then, the proposed approach, the Kmeans method and the EM method calculate the pixel mean values of different pixel classes from the de-noised fuzzy images. The errors are computed by the following formulation:

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Er  0  0  1  1  2  2  3  3 (21) where  0 , 1 ,  2 and 3 are the computed pixel mean values by the proposed approach, the EM method or the K-means method. We plot the average errors for the 100 synthesized images by the EM method, K-means method and the proposed approach in Fig. 19. As a whole, the plotted errors of EM and K-means methods are above the plotted error of the proposed approach. The quantitative comparison is shown in Table. 2. As can be seen, the proposed approach is significantly more accurate than EM and K-means in calculating means for different pixel classes during image segmentation. For the quantitative comparison of the segmentation accuracy, we first compared the proposed approach, Kernel graph cuts, Normalized graph cuts, Graph, Mean-shift and K-means in Table. 3 with two measures: the probabilistic rand index (Unnikrishnan et al., 2005) and variation of information (Yang et al., 2008) using the Berkeley dataset. The probabilistic rand index is called PRI and the higher the better. The variation of information is

Fig.19. Comparison of the errors in finding the pixel means

Fig.20. Comparison of the F-measures computed by different methods for the first synthesized image set (LR-AC represents localized region based active contour; NE-AC represents active contour without edge; DR-LS represents distance regularized level set.)

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(a) (b) Fig. 22. Demonstration of groundtruth boundary overlaying on the synthesized fuzzy image with Gaussian noise magnitude equal 10. (a) Groundtruth boundary for the first synthesized image set; (b) Groundtruth boundary for the second synthesized image set.

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Fig.21. Comparison of the F-measures computed by different methods for the second synthesized image set (LR-AC represents localized region based active contour; NE-AC represents active contour without edge; DR-LS represents distance regularized level set.)

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Table. 3 Comparison of the proposed approach with state of the art methods Graph Mean-shift Proposed Kernel graph cuts N- cuts 0.7984 0.7650 0.7229 0.7841 0.7555 2.2247 2.4091 2.9329 2.6647 2. 477

K-means 0.7882 2. 304

Table. 4 Comparison of the average F-measure computed by different methods from the first synthesized image Region based Active contours Distance regularized GM EM K-means FFCM Proposed active contour without edge level set 0.9860 0.8937 0.9482 0.9385 0.7893 0.9277 0.8051 0.8001

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Table. 2 Comparison of the proposed approach with EM and K-means Methods Average error Average time EM 7.8360 1.07 s K-means 7.8177 0.11 s Proposed 5.7431 0.12 s

Table. 5 Comparison of the average F-measure computed by different methods from the second synthesized image Region based Active contours Distance regularized GM EM K-means FFCM Proposed active contour without edge level set 0.9801 0.9369 0.9617 0.9015 0.4015 0.9438 0.7038 0.5080 F-measure

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Image segmentation is important for many computer vision applications and thus a lot of efforts have been put into deg image segmentation algorithms to solve specific problems. After decades of evolution, the segmentation methods converge to several categories, e.g. graph based methods (Felzenszwalb and Huttenlocher, 2004; Shi and Malik, 2000; Salah et al., 2011), stochastic modeling based methods (Carson et al., 2002; Won and Gray, 2004; Jain, 2010; Wang and Zhang, 2012) and variational formulations based methods (Chan and Vese, 2001; Li et al., 2010; Lankton and Tannenbaum, 2008; Arbelaez et al., 2011). Unfortunately, these methods are only good at segmenting some specific types of images while perform poorly in other types of images. As a result, it is necessary to exploit new theory or directions to come up with more effective methods that could segment more classes of images robustly. Image segmentation remains challenging because the vast variety of images are usually with different properties. For instance, the histogram distributions of different types of images might comply with different mathematic formulations, although they are assumed to be Gaussian distributed in most cases. The noise and the overlapping of different pixel classes in the histogram distribution make the estimation of the parameters challenging. Furthermore, the inhomogeneous spatial scale of different regions makes the segmentation more difficult. In general, the image has local properties and global properties that should be utilized effectively during segmentation. EM and Kmeans have been served in estimating the global parameters in many image segmentation applications. Unfortunately, their accuracies are limited by the complexity and variety of the image

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histogram distribution and the noise distribution. In this paper, we propose an approach that utilizes the slope difference distribution of the image histogram to calculate the means for different pixel classes. The proposed approach could effectively deal with the noise and overlaps of multiple distributions. As it turned out that the slope difference distribution method is more accurate than EM and K-means methods. For the local property of the image, we utilize the Gibbs distribution to characterize its distribution. We utilize an iterative gradient descent method to minimize the Gibbs energy in the image. To refine the boundary, we deform the extracted boundary to the nearest Canny edges and use polynomial splines to achieve the global smoothness. In nature, the proposed approach is different from the traditional categories of state of the art segmentation methods. In performance, the proposed approach is more robust than state of the art methods because it utilizes the properties of the image more generally. 5. Conclusion In this paper, a new direction of image segmentation based on both the local property and the global property of the image is exploited. The proposed approach calculates the means of different pixel classes from the slope difference distribution and it is significantly more accurate than the benchmark EM and Kmeans methods. The image is labelled by clustering the pixel to the nearest means. To improve the accuracy of the labelled result, we suppress the inhomogeneity inside the image by minimizing the Gibbs energy function. To improve the accuracy of the extracted boundary, we deform the boundary to the nearest Canny edges and then optimize the whole boundary with

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Acknowledgments The author would like to thank the Chinese Academy of Sciences for the funding with the grant number Y5A1270101.

Wang, Z.Z., Zhang, Y.M., 2012 Segmentation of fuzzy images: a novel and fast two-step pseudo MAP method, Machine Vision and Applications, 23(6), 1209-1218. Won, C.S., Gray, R.M., 2004 Stochastic image processing. Information technology: Transmission, Processing, and Storage. Yang, A.Y., Wright, J., Ma, Y., Sastry, S.S., 2008 Unsupervised segmentation of natural images via lossy data compression, Computer Vision and Image Understanding, 110, 212–225.

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