Applied Computational Intelligence and Soft Computing

Volume 2016, Article ID 8508329, 14 pages

http://dx.doi.org/10.1155/2016/8508329

## Semisupervised Soft Mumford-Shah Model for MRI Brain Image Segmentation

^{1}School of Information Science & Engineering, Changzhou University, Changzhou 213164, China^{2}Department of Natural Science & Mathematics, West Liberty University, West Liberty, WV 26074, USA

Received 27 March 2016; Accepted 29 May 2016

Academic Editor: Serafín Moral

Copyright © 2016 Hong-Yuan Wang and Fuhua Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

One challenge of unsupervised MRI brain image segmentation is the central gray matter due to the faint contrast with respect to the surrounding white matter. In this paper, the necessity of supervised image segmentation is addressed, and a soft Mumford-Shah model is introduced. Then, a framework of semisupervised image segmentation based on soft Mumford-Shah model is developed. The main contribution of this paper lies in the development a framework of a semisupervised soft image segmentation using both Bayesian principle and the principle of soft image segmentation. The developed framework classifies pixels using a semisupervised and interactive way, where the class of a pixel is not only determined by its features but also determined by its distance from those known regions. The developed semisupervised soft segmentation model turns out to be an extension of the unsupervised soft Mumford-Shah model. The framework is then applied to MRI brain image segmentation. Experimental results demonstrate that the developed framework outperforms the state-of-the-art methods of unsupervised segmentation. The new method can produce segmentation as precise as required.

#### 1. Introduction

In recent years, MRI based medical image processing and analysis have been studied widely. Among these researches, segmentation is at the first stage and is fundamental for poster processing and analysis. One of the most important applications in medical image processing is MRI brain image segmentation. It has been noticed that, by calculating changes of volumes of different brain tissues (called white matter, gray matter, and cerebrospinal fluid in image processing), some brain related diseases can be found at their early stage [1]. However, there are two challenges in calculating the volumes of different matters in MRI brain images. One challenge is the calculation of partial volumes appearing usually at the border of different tissues, due to limited resolution [2–5]; another challenge is the segmentation of central gray matter due to the faint contrast with respect to the surrounding white matter [6]. This paper addresses the later challenge.

Central gray matter lies in the central area of brains. Its intensity is usually very close to the white matter located not in central areas. In detail, the intensity of central gray matter is a little smaller than the intensity of white matter located in central area, but usually very close to or even greater than the intensity of this white matter near the outer layer. As a result, it is deficient for intensity based unsupervised segmentation methods in distinguishing central gray matter from white matter for MRI brain images.

In general, unsupervised methods explore the intrinsic data features to partition an image into regions with different statistics. The segmentation procedure can be implemented using some assigned algorithm automatically without human beings’ interaction or interfering. There are several cases that unsupervised methods either fail to work or are deficient. One case is of intensity inhomogeneity. Another case is when some parts of different classes have almost the same intensities or features. In the first case, it is solved by bias correction methods [7, 8] or by stochastic methods [9, 10] or both [6]. When bias correction methods are used, the bias field is always assumed smooth. When a bias field is not smooth, an alternative way is to use stochastic methods treating pixel intensities as randomly distributed random variables. However, stochastic methods in dealing with bias only work well when the bias is not strong. Nevertheless, some traditional unsupervised methods can still work for the first case (with bias). However, unsupervised segmentation methods usually fail to work efficiently for the second case.

Different from unsupervised segmentation, supervised image segmentation is a technique to partition an image using either known images or known features of some parts of the image to direct the segmentation.

Machine-learning based image segmentation is such a method that uses a collection of known images having the same type as a given image (to be segmented) to direct the image segmentation [11, 12]. The direction is implemented by a learning mechanic. The machine-leaning based image segmentation methods are originated from general classification methods. Such methods, when applied to image segmentation, deal with each pixel as an isolated object without considering the relation to its neighbors such as smoothness of the intensities inside a class. Moreover, the methods are usually based on algorithms, not based on a mathematical model, and therefore mathematically less precise.

Another way for supervised segmentation is to use some patches of a given image to direct the image segmentation. It assigns some regions for each class in advance based on prior knowledge and then uses the features of the known regions as constraints to model image segmentation. A class of such methods is supervised image matting [13–15]. Image matting studies the problem of accurate foreground estimation in images and videos. It is essentially a two-phase image segmentation and usually deals with natural images that are very complicated. During image matting, supervised methods are usually used by assigning some regions as foreground and then use the assigned regions as reference to help extract the foreground. Image matting also provides interactive segmentation. Interactive method is also discussed in the famous Grabcut [16] method which deals with an image as a graph under discrete settings.

There are two shortcomings when using supervised image matting for image segmentation. First, image matting works only for images with two classes and assumes that the image is a linear combination of background and foreground. Second, there is no theoretical proof addressing why a supervised or interactive matting method is more reliable than an unsupervised image matting. Results are claimed better only based on visual effect.

Mumford-Shah model is a multiphase image segmentation model that has been extensively investigated [10, 17–22]. Original Mumford-Shah model assumes an image to be a piecewise smooth function [23]. Later researches usually assume each piece of the function to have some special property such as piecewise polynomial [24]. Most often, the image in a Mumford-Shah model is assumed to be piecewise constant [18–20]. In the later case, the model is sometimes implemented under the assumption that the mean of each class is known based on prior knowledge. Since the assumption of piecewise constant is too strong and may limit its application, some varied forms of Mumford-Shah model are also developed [8, 22, 25].

Considering that soft segmentation model is usually more flexible and makes it possible to produce a globally optimized result, Jianhong Shen extended Mumford-Shah model for soft segmentation [10], where each pixel can partly belong to more than one class. Membership functions are used in the model to denote the percentage or probability that a pixel belongs to each class. The value of a membership function at some pixel can be viewed as either the probability of the pixel belonging to the corresponding class such as fuzzy segmentation model [26, 27] or the percentage of the pixel belonging to the corresponding class such as partial volume segmentation [5, 28–30].

In this paper, a soft version of the piecewise constant Mumford-Shah model is introduced. Then, a frame work of semisupervised and interactive image segmentation is developed based on the soft piecewise constant Mumford-Shah model using Bayesian principle. The developed model is proved to be an extension of the general unsupervised soft Mumford-Shah model. The semisupervised and interactive framework can produce segmentation result as precise as required. The rest of the paper is organized as below. Section 2 addresses the importance of supervised segmentation methods and its basic idea. For a given synthetic image, different segmentation results are presented when different methods, an unsupervised method, and a supervised method are used. Section 3 introduces the development of the proposed framework. Section 4 presents the numerical analysis and algorithm implementation. The efficiency of the framework is shown in Section 5 using experiments, where the application to MRI brain images is especially introduced. Finally, some comments, conclusion, and future work are addressed in Section 6.

#### 2. Introduction to Semisupervised Segmentation

Unsupervised image segmentation utilizes the inherent image features to partition an image into different classes such that the pixels in the same class share the same or similar features while pixels in different classes have quite different features. The lowest level image feature is image intensity. Most of the unsupervised image segmentation models directly use image intensities to classify pixels. The advantage of unsupervised image segmentation is well-known. For example, it is fast; it does not need human’s interaction; even the number of classes is not required to be known before implementation. Meanwhile, the disadvantages are also well-known. For example, in Mumford-Shah model, if the number of classes is unknown, it is hard to give an expected result: different numbers of classes will lead to different segmentation results. Another example is the initialization during the implementation for a nonconvex model. When a model is nonconvex, the implementation usually leads to a local minimizer that may not be the expected result. There are also other shortcomings for unsupervised image segmentation. Some shortcomings can be or has been resolved by new mathematical methods. For example, when the number of classes is unknown, Chiu [31] and Wang [32] proposed different ways to solve the problem under an unsupervised setting. However, some drawback of unsupervised image segmentation could not be solved due to the inherent features of digital images. For example, when the pixels in the same class have quite different intensities or pixels in different classes have very close intensities, it is generally impossible to achieve an ideal result using an unsupervised segmentation model that is based on intensities only (this statement is not really true for texture image segmentation). Figure 1 shows the difference at an extreme case.