Image Segmentation Based on Maximum Entropy and Kernel Self-Organizing Map

LIN Chang, YU Chong-xu State Key Laboratory of Information Photonics and Optical Communications

Beijing University of Posts and Telecommunications Beijing, China

Abstract—This paper proposes a segmentation method based on information theory. The entropy of the image is regarded as the objective function to be optimized. It is maximized during the segmenting process. At first, the kernel self-organizing map is applied to cluster the input vectors of the image into groups according to their attributes, and it keeps entropy maximization meanwhile. Then the clustering result is partitioned using the maximum entropy principle. Finally, the image is segmented according to the partition of clusters. Objective evaluation methods are applied to assess the performance of the method. Experimental results show that this method has the advantages of fault tolerance and adaptability, and it can separate salient objects from the background correctly.

Keywords-Image segmentation; Maximum entropy; Kernel method; Self-organizing map

I. INTRODUCTION Image segmentation is defined as the process of partitioning

the image into different sub regions of homogeneity. It is a fundamental processing step of image analysis and image understanding. The goal of image segmentation is to cluster pixels into salient image regions. Of various segmentation methods, clustering is one of the most important and widely used approaches. Clustering is a process of organizing the objects into groups based on their attributes. The most popular clustering methods are K-means and Fuzzy C-means (FCM) algorithms. K-Means is efficient, but its number of clusters must be supplied as a parameter. FCM is an improvement of K-Means on the basis of fuzzy mathematics. It is more suited to the uncertainty and ambiguity characteristics of the image. However, FCM is extremely sensitive to initial parameters. It doesn’t take advantage of spatial information, and is susceptible to noise.

Compared with these classical methods, the self-organizing map (SOM) based segmentation approach has the advantage of self-learning, fault tolerance, and adaptability. It can form arbitrarily complex clusters, preserve the topology of the original input data set, provide a natural measure for the distance of a point from a cluster, and produce clusters that better match the desired classes. These lead to a good segmentation results [1]. SOM algorithm is a powerful tool for exploring large amounts of high-dimensional data. However it has two fundamental limitations: (1) the estimate of probability density function of the input space provided by the algorithm

lacks accuracy, and (2) the formulation of the algorithm has no objective function that could be optimized [2]. These make the process analysis and objective optimization become very difficult, and the algorithm may not meet the purpose of image segmentation completely.

The entropy is a measure of the amount of information conveyed by message. The maximum entropy principle has answered the question of how to choose an optimum probability model of a stochastic system with a set of known states [3]. It is confirmed that the maximum entropy will provide the best probability estimation of the image [4]. In this paper, we propose a segmentation method based on maximum entropy and Van Hulle’s kernel feature-mapping model [5]. The entropy of the image is used as the objective function to be optimized. Our purpose is to maximize the entropy during the segmenting process. The kernel self-organizing map algorithm is applied to cluster the input vectors and keep entropy maximization. Then the clustering result is partitioned using the maximum entropy principle. Many images are used to test the performance of our method. Some objective evaluation methods [6-9] are applied to assess the quality of the segmentation results. Finally, we conclude the paper and discuss the future work.

II. SEGMENTATION BASED ON MAXIMUM ENTROPY

A. The Maximum Entropy Principle Consider a discrete random variable },2,1{ nkxX k …== ,

let the event kxX = occur with probability kp , then the amount of information gained after observing the event kxX = is:

kk

pp

log)1log( −= , where )log(• is the base-2 logarithm

function. Define ∑=

−=n

kkk ppXH

1

log)( , )(XH is called the

entropy of random variable X . For continuous random variable X with the probability density function )(xpX ,

similarly define ∫∞

∞−−= dxxpxpXh XX )(log)()( , )(Xh is called

the differential entropy of X .

The entropy is a measure of the average amount of information conveyed per message. That is, the more entropy

978-1-4577-1964-6/12/$31.00 ©2012 IEEE

an event has, the more information it contains. The maximum entropy principle indicates:

When an inference is made on the basis of incomplete information, it should be drawn from the probability distribution that maximizes the entropy, subject to constraints on the distribution [3].

B. Van Hulle’s Kernel Self-Organizing Map To improve the density estimation properties and the noise

tolerance of the SOM algorithm, Van Hulle developed a new learning algorithm for kernel-based topographic map formation that was aimed at maximizing the map’s joint entropy [5]. This is achieved by maximizing the differential entropies of the kernel outputs individually and minimizing the mutual information between the kernel outputs. The latter is achieved heuristically by having a competitive stage in the learning process.

In a kernel SOM, each neuron in the lattice structure of the map acts as a kernel. Let the random variable iY refer to the output of the ith kernel, its probability density function is denoted by )( iY yP

i, and iy refers to a sample value of iY . The

differential entropy of iY is defined by

∫∞

∞−−= iiYiYi dyypypYH

ii)(log)()( (1)

Let the kernel be denoted by ),,( iiwxk σ , where x is the input vector of dimensionality d, iw is the weight vector of the ith kernel, and iσ is its width; the index i=1,2,…N, where N is the number of neurons constituting the lattice structure, then

Niwxky iii ,,2,1),,,( …== σ (2)

Since the kernel is radial symmetric, we have ),(),,( iiii wxkwxk σσ −= , where iwx − is the Euclidean

distance between x and iw .

Suppose iy has “bounded” support, according to the properties of entropy, )( iYH will be maximized when iY is uniformly distributed. This occurs when the output distribution matches the cumulative distribution function of the input space. For a Gaussian-distributed input vector x, Van Hulle found that the cumulative distribution function of

iwx − was the incomplete gamma distribution, and deduced that the corresponding kernel had the definition as follows:

Nid

wxd

wxk i

i

ii ,,2,1,)

2(

)2

,2

(),,(

2

2

…=Γ

−Γ

= σσ (3)

Where )(•Γ is the gamma function.

After the kernel function has defined, the algorithm for self-organized topographic formation is formulated by deriving formulas for the gradients of the )( iYH with respect to the kernel parameters iw and iσ . The formulas of the weight and the kernel-width adjustment are obtained as follows:

2i

iwi

wxwσ

η −=Δ , (4)

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

−=Δ 12

2

i

i

ii d

wxσσ

ησ σ , (5)

with wη the learning rate of iw and ση the learning rate of iσ .

To maintain the statistically independence between the kernel outputs, Van Hulle put the mechanism of kernel adaptation in the competitive learning framework just as the SOM algorithm.

Finally, the steps of kernel self-organizing map can be summarized as:

(1) Choose random values to initial iw and iσ .

(2) Draw a sample x from the input space with a certain probability.

(3) Identify the winning neuron i(x), using the criterion: )(maxarg)(* xyxi i= , i=1,2,…N.

(4) Adjust the weight vector and width of kernels which lie inside the topological neighborhood of i(x) by using the formulas (4) and (5). The neighborhood function is

defined as: )2

exp(),( 2

2

* *

Λ

−−=Λ

σii rr

ii here, with Λσ the

neighborhood function range, and ir neuron i’s lattice coordinate.

(5) Continue with step 2 until no noticeable changes in the adjustment are observed.

C. Segmentation Method Base on Van Hulle’s Kernel Self-Organizing Map and Maximum Entropy The purpose of maximum entropy principle is to answer the

question of how to choose an optimum probability model of a stochastic system with a set of known states. The maximum entropy will provide the best probability estimation of the image. This has been clearly verified since T. Pun’s work. In [4], T. Pun presented an “entropic thresholding” approach to successfully solve the segmentation problem when the histogram is not clearly bimodal. However, the calculation of entropy is very time-consuming, especially for color images which always contain a large amount of information. This

makes segmenting in the color space directly using maximum entropy almost impossible. To overcome this problem, we apply Van Huller’s algorithm to cluster the colors to reduce the complexity of data. This algorithm provides a good approximation to the input space. The entropy of the image is kept maximization during the clustering process. As a result, when partitioning the image in its feature space using maximum entropy, the corresponding segmentation in the original space will also satisfy the maximum entropy principle. Now we can describe our segmentation method as follows:

(1) Transform the RGB data into HUV values, map the HUV vectors to the feature space using SOM algorithm, and separate the foreground and background from the image according to clustering result of the H component. This step aims at reducing the influence of illumination. It also is used to detect the dominant colors in the image.

(2) Base on the above coarse separation result, apply the kernel self-organizing algorithm to map the RGB vectors to the feature space, partition the feature vectors using maximum entropy principle. This is the main step of the method.

(3) Segment the image according to the partitioning result in the feature space.

(4) Merge the small regions with one of their surrounding regions which color distance is closest, if necessary.

III. EXPERIMENTAL RESULTS Image segmentation is a fundamental process of image

processing, but it also is a complex problem with no exact

solution. Although innumerable image segmentation methods have been proposed, thus far there still lacks a reliable objective evaluation method to assess the quality of the segmentation. In recent years, a few objective evaluation methods have been proposed [6-9], but these methods are based mainly on empirical analysis or have no enough theoretical grounding. So in our experiments, segmentation performance has to be evaluated by subjectively judging several sample images as usually. However, we try to make it as objectively as possible by using the research results of the previous studies.

Fig.1 shows the segmentation result of image puppies which is a 1024×768 true colors image. First, the number of colors is reduced to 9 through clustering, then the clustering result is partitioned using maximum entropy, finally the image is segmented into ‘color and spatial homogeneous’ regions, as shown in Fig.1 (b) and (c). Fig.2 is the segmentation experiment of image 12003 which comes from the Berkeley segmentation database BSDS500 [6,7]. Fig.2 (b) is the segmentation result obtained by our approach. Fig.2 (c) is the manually segmented result provided by BSDS500. From Fig.1 and Fig.2 we can see that our method can extract the salient object from the image correctly. Ge et al. believed that the capability to separate salient objects from the background is a general measure for evaluating performance [8]. According to this viewpoint, our method can obtain ‘accurate’ segmentations.

Fig.3 (a) and (c) also come from BSDS500. Fig.3 (b) is our segmentation result. It shows that the obtained boundaries are very accurate, which are comparable with the ground-true segmentation.

(a) original image (b) color clustering (c) segmentation result

Fig.1 Segmentation experiment of image puppies

(a) original image (b) our segmentation (c) ground-truth segmentation

Fig.2 Segmentation experiment of image 12003

(a) original image (b) our segmentation (c) ground-truth segmentation

Fig.3 Segmentation experiment of image 42049

IV. CONCLUSIONS AND FUTURE WORK When employed as a data clustering technique, the SOM

algorithm will provide a good approximation to the input space. Van Hulle used the joint entropy of kernel outputs as the object function to derive a new learning algorithm which was aimed at improving the density estimation properties. We find that the object function of this algorithm is consistent with the goal of image segmentation. On this basis, we propose a maximum-entropy segmentation method based on this algorithm. Many images are applied to test the performance of our method. Experimental results show that it can separate targets from the background correctly, and obtain accurate edges. It also has the advantages of fault tolerance and adaptability. Next we intend to analysis this method in detail, and find out a quantitative evaluation technique to assess its performance objectively.

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