Fuzzy Measures and Weighted Co-Occurrence Matrix for Segmentation of Brain MR Images

, T1, and T2. Therefore, majority of research in medical image analysis concerns the MR images [36].


Conventionally, these images are interpreted visually and qualitatively by radiologists. Advanced research requires quantitative information such as the size of the brain ventricles after a traumatic brain injury or the relative volume of ventricles to brain. Fully automatic methods sometimes fail, producing incorrect results and requiring the intervention of a human operator. This is often true due to restrictions imposed by image acquisition, pathology, and biological variation. Hence, it is important to have a faithful method to measure various structures in the brain. One of such methods is the segmentation of images to isolate objects and regions of interest.

Image segmentation is an indispensable process in the visualization of human tissues, particularly during clinical analysis of medical images. In the analysis of medical images for computer-aided diagnosis and therapy, segmentation is often required as a preliminary stage. The success of an image analysis system depends on the quality of segmentation [32, 33, 36]. Medical image segmentation is a complex and challenging task due to the intrinsic nature of the images. The brain has a particularly complicated structure and its precise segmentation is very important for detecting tumors, edema, and necrotic tissues, in order to prescribe appropriate therapy [32, 33, 36].

Segmentation is a process of partitioning an image space into some nonoverlapping meaningful homogeneous regions. If the domain of the image is given by $$\Omega $$, then the segmentation problem is to determine the sets $$S_k \subset \Omega $$, whose union is the entire domain $$\Omega $$. The sets that make up a segmentation must satisfy


$$\begin{aligned} \Omega =\bigcup _{k=1}^k S_k, \end{aligned}$$

(11.1)
where $$S_k \cap S_j =\emptyset $$ for $$k \ne j$$, and each $$S_k$$ is connected. Hence, a segmentation method is supposed to find those sets that correspond to distinct anatomical structures or regions of interest in the image.

Many image processing techniques have been proposed for the MR image segmentation [2, 7], most notably thresholding [10, 14, 34], region-growing [18], edge detection [35], pixel classification [25, 31], and clustering [1, 11, 40]. Some algorithms using the neural network approach have been investigated in the MR image segmentation problems [5, 6]. The segmentation of the MR images using fuzzy $$c$$-means has been reported in [1, 4, 6, 12, 28, 45]. Image segmentation using rough sets has also been done in [8, 16, 17, 19, 30, 42–44]. Recently, a review is reported in [9] on the application of rough sets and near sets in medical imaging.

Thresholding is one of the old, simple, and popular techniques for image segmentation. It can be done based on global (for example, gray level histogram of the entire image) or local information (for example, co-occurrence matrix) extracted from the image. A series of algorithms for image segmentation based on histogram thresholding can be found in the literature [2, 7, 10, 14, 20, 25, 34, 37]. Entropy-based thresholding algorithms have been proposed in [21–23, 29]. One of the main problems in medical image segmentation is uncertainty. Some of its sources include imprecision in computations and vagueness in class definitions. In this background, the possibility concept introduced by the fuzzy set theory has gained popularity in modeling and propagating uncertainty in medical imaging applications [15, 16]. Also, since the fuzzy set theory is a powerful tool to deal with linguistic concepts such as similarity, several segmentation algorithms based on fuzzy set theory are reported in the literature [3, 13, 26, 29, 38].

In general, all histogram thresholding techniques based on fuzzy set theory work very well when the image gray level histogram is bimodal or multimodal. On the other hand, a great deal of medical images are usually unimodal, where the conventional histogram thresholding techniques perform poorly or even fail. In this class of histograms, unlike the bimodal case, there is no clear separation between object and background pixel occurrences. Hence, to find a reliable threshold, some adequate criteria for splitting the image histogram should be used. In [38], an approach to threshold the histogram according to the similarity between gray levels has been proposed.

This chapter presents a new algorithm, termed as the FMWCM [13, 14], to threshold the image histogram. It is based on a fuzzy measure and the concept of weighted co-occurrence matrix. The second order fuzzy measures such as fuzzy correlation, fuzzy entropy, and index of fuzziness, are used for assessing such a concept. The local information of the given image is extracted through a modified co-occurrence matrix. The FMWCM technique consists of two linguistic variables {bright, dark} modeled by two fuzzy subsets and a fuzzy region on the gray level histogram. Each of the gray levels of the fuzzy region is assigned to both defined subsets one by one and the second order fuzzy measure using weighted co-occurrence matrix is calculated. The ambiguity of each gray level is determined from the fuzzy measures of two fuzzy subsets. Finally, the strength of ambiguity for each gray level is computed. The multiple thresholds of the image histogram are determined according to the strength of ambiguity of the gray levels using a nearest mean classifier. Experimental results reported in this chapter confirm that the FMWCM method is robust in segmenting brain MR images compared to existing popular thresholding techniques.

The rest of this chapter is as follows: In Sect. 11.2, some basic definitions about fuzzy sets and second order fuzzy measures along with co-occurrence matrix are reported. The FMWCM algorithm for histogram thresholding is presented in Sect. 11.3. Experimental results and a comparison with other thresholding methods are presented in Sect. 11.4. Concluding remarks are given in Sect. 11.5.



11.2 Fuzzy Measures and Co-Occurrence Matrix


This section presents the basic notions in the theory of fuzzy sets and the concept of co-occurrence matrix, along with different second order fuzzy measures and fuzzy membership function.


11.2.1 Fuzzy Set


A fuzzy subset $$A$$ of the universe $$X$$ is defined as a collection of ordered pairs


$$\begin{aligned} A = \{(\mu _A(x), x), \forall x \in X \} \end{aligned}$$

(11.2)
where $$\mu _A(x)$$ denotes the degree of belonging of the element $$x$$ to the fuzzy set $$A$$ and $$0\le \mu _A(x) \le 1$$. The support of fuzzy set $$A$$ is the crisp set that contains all the elements of $$X$$ that have a nonzero membership value in $$A$$ [46].

Let $$X=[x_{mn}]$$ be an image of size $$M\times N$$ and $$L$$ gray levels, where $$x_{mn}$$ is the gray value at location $$(m,n)$$ in $$X$$, $$x_{mn} \in G_L$$, $$G_L=\{0,1,2,.....,L-1\}$$ is the set of the gray levels, $$m=0,1,2,\cdots , M-1$$, $$n=0,1,2,\cdots , N-1$$, and $$\mu _X(x_{mn})$$ be the value of the membership function in the unit interval $$[0,1]$$, which represents the degree of possessing some brightness property $$\mu _X(x_{mn})$$ by the pixel intensity $$x_{mn}$$. By mapping an image $$X$$ from $$x_{mn}$$ into $$\mu _X(x_{mn})$$, the image set $$X$$ can be written as


$$\begin{aligned} X =\{\mu _X(x_{mn}), x_{mn}\}. \end{aligned}$$

(11.3)
Then, $$X$$ can be viewed as a characteristic function and $$\mu _X$$ is a weighting coefficient that reflects the ambiguity in $$X$$. A function mapping all the elements in a crisp set into real numbers in $$[0,1]$$ is called a membership function. The larger value of the membership function represents the higher degree of the membership. It means how closely an element resembles an ideal element. Membership functions can represent the uncertainty using some particular functions. These functions transform the linguistic variables into numerical calculations by setting some parameters. The fuzzy decisions can then be made. The standard $$S$$-function, that is, $$S(x_{mn}; a,b,c)$$, of Zadeh is as follows [46]:


$$\begin{aligned} \mu _X(x_{mn}) = \left\{ \begin{array}{ll} 0 &{} x_{mn} \le a\\ 2 \left[ \frac{x_{mn}-a}{c-a} \right] ^2 &{} a \le x_{mn} \le b\\ 1-2 \left[ \frac{x_{mn}-c}{c-a} \right] ^2 &{} b \le x_{mn} \le c\\ 1 &{} x_{mn} \ge c\\ \end{array} \right. \end{aligned}$$

(11.4)
where $$b=\frac{(a+c)}{2}$$ is the crossover point for which the membership value is 0.5. The shape of $$S$$-function is manipulated by the parameters $$a$$ and $$c$$.


11.2.2 Co-Occurrence Matrix


The co-occurrence matrix or the transition matrix of the image $$X$$ is an $$L\times L$$ dimensional matrix that gives an idea about the transition of intensity between adjacent pixels. In other words, the $$(i,j)$$th entry of the matrix gives the number of times the gray level $$j$$ follows the gray level $$i$$, that is, the gray level $$j$$ is an adjacent neighbor of the gray level $$i$$, in a specific fashion. Let $$a$$ be the $$(m,n)$$th pixel in $$X$$ and $$b$$ denotes one of the eight neighboring pixels of $$a$$, that is,


$$\begin{aligned} \begin{array}{ll} b\in a_8 &{}= \left\{ (m,n-1),(m,n+1),(m+1,n),(m-1,n),(m-1, n-1), \right. \\ &{}\qquad \left. (m-1,n+1),(m+1,n-1),(m+1,n+1) \right\} \end{array}\end{aligned}$$



$$\begin{aligned} {\text {then}}~~t_{ij}=\sum _{\begin{array}{c} a\in X \\ b\in a_{8} \end{array}} \delta ; \end{aligned}$$

(11.5)



$$\begin{aligned} \text {where}~~ \delta = \left\{ \begin{array}{ll} 1 &{} \text {if gray level value of} \;a \;\text {is}\; i \;\text {and that of} \;b \;\text {is} \;j\\ 0 &{} \text {otherwise.} \end{array} \right. \end{aligned}$$

(11.6)
Obviously, $$t_{ij}$$ gives the number of times the gray level $$j$$ follows gray level $$i$$ in any one of the eight directions. The matrix $$T=[t_{ij}]_{L\times L}$$ is, therefore, the co-occurrence matrix of the image $$X$$.


11.2.3 Second Order Fuzzy Correlation


The correlation between two local properties $$\mu _1$$ and $$\mu _2$$ (for example, edginess, blurredness, and texture) can be expressed in the following ways [27]:


$$\begin{aligned} C(\mu _1,\mu _2)=1-\frac{\displaystyle 4\sum _{i=1}^{L} \sum _{j=1}^{L}[\mu _1(i,j)-\mu _2(i,j)]^2 t_{ij}}{Y_1+Y_2} \end{aligned}$$

(11.7)
where $$t_{ij}$$ is the frequency of occurrence of the gray level $$i$$ followed by $$j$$, that is, $$T=[t_{ij}]_{L\times L}$$ is the co-occurrence matrix defined earlier, and


$$\begin{aligned} Y_k=\sum _{i=1}^{L} \sum _{j=1}^{L} [2\mu _k(i,j)-1]^2 t_{ij};~~k=1,2. \end{aligned}$$

(11.8)
To calculate the correlation between a gray-tone image and its two-tone version, $$\mu _2$$ is considered as the nearest two-tone version of $$\mu _1$$, that is,


$$\begin{aligned} \mu _2(x) = \left\{ \begin{array}{ll} 0 &{} \text {if}\; \mu _1(x) \le 0.5\\ 1 &{} \text {otherwise.}\\ \end{array} \right. \end{aligned}$$

(11.9)


11.2.4 Second Order Fuzzy Entropy


Out of the $$n$$ pixels of the image $$X$$, consider a combination of $$r$$ elements. Let $$S_i^r$$ be the $$i$$th such combination and $$\mu (S_i^r)$$ denotes the degree to which the combination $$S_i^r$$, as a whole, possesses the property $$\mu $$. There are $$\left( \begin{array}{c} n \\ r \end{array} \right) $$ such combinations. The entropy of order $$r$$ of the image $$X$$ is defined as [24]


$$\begin{aligned} H^{(r)}=-\frac{1}{N} \sum _{i=1}^{N} \left[ \mu (S_i^r)\mathrm{{ ln}}\{\mu (S_i^r)\}+ \{1-\mu (S_i^r)\}\mathrm{{ln}}\{1-\mu (S_i^r)\}\right] \end{aligned}$$

(11.10)
with logarithmic gain function and $$N=\left( \begin{array}{c} n \\ r \end{array} \right) $$. It provides a measure of the average amount of difficulty or ambiguity in making a decision on any subset of $$r$$ elements as regards to its possession of an imprecise property. Normally, these $$r$$ pixels are chosen as adjacent pixels. For the present investigation, the value of $$r$$ is chosen as 2.


11.2.5 Second Order Index of Fuzziness


The quadratic index of fuzziness of an image $$X$$ of size $$M \times N$$ reflects the average amount of ambiguity or fuzziness present in it by measuring the distance (quadratic) between its fuzzy property plane $$\mu _1$$ and the nearest two-tone version $$\mu _2$$. In other words, the distance between the gray-tone image and its nearest two-tone version [29]. If we consider spatial information in the membership function, then the index of fuzziness takes the form


$$\begin{aligned} I(\mu _1,\mu _2)=\frac{\displaystyle 2 \left\{ \sum _{i=1}^{L} \sum _{j=1}^{L}[\mu _1(i,j)-\mu _2(i,j)]^2 t_{ij} \right\} ^{\frac{1}{2}}}{\sqrt{MN}} \end{aligned}$$

(11.11)
where $$t_{ij}$$ is the frequency of occurrence of the gray level $$i$$ followed by $$j$$.

For computing the second order fuzzy measures such as correlation, entropy, and index of fuzziness of an image, represented by a fuzzy set, one needs to choose two pixels at a time and to assign a composite membership value to them. Normally these two pixels are chosen as adjacent pixels.


11.2.6 2D S-Type Membership Function


This section presents a two dimensional $$S$$-type membership function that represents fuzzy bright image plane assuming higher gray value corresponds to object region. The 2D $$S$$-type membership function reported in [28] assigns a composite membership value to a pair of adjacent pixels as follows: For a particular threshold $$b$$,

1.

($$b,b$$) is the most ambiguous point, that is, the boundary between object and background. Therefore, its membership value for the fuzzy bright image plane is 0.5.

 

2.

If one object pixel is followed by another object pixel, then its degree of belonging to object region is greater than 0.5. The membership value increases with increase in pixel intensity.

 

3.

If one object pixel is followed by one background pixel or vice versa, the membership value is less than or equal to 0.5, depending on the deviation from the boundary point ($$b,b$$).

 

4.

If one background pixel is followed by another background pixel, then its degree of belonging to object region is less than 0.5. The membership value decreases with decrease of pixel intensity.

 
Instead of using fixed bandwidth ($$\Delta b$$), the parameters of $$S$$-type membership function are taken as follows [38]:


$$\begin{aligned} b = \frac{\displaystyle \sum _{i=p}^{q} x_i \cdot h(x_i)}{\displaystyle \sum _{i=p}^{q} h(x_i)}; \end{aligned}$$

(11.12)



$$\begin{aligned} \Delta b = \max \{|b-(x_i)_{\text {min}}|,|b-(x_i)_{\text {max}}|\}; \end{aligned}$$

(11.13)



$$\begin{aligned} c = b + \Delta b; \end{aligned}$$

(11.14)



$$\begin{aligned} a = b - \Delta b; \end{aligned}$$

(11.15)
where $$h(x_i)$$ denotes the image histogram, and $$x_p$$ and $$x_q$$ are the limits of the subset being considered. The quantities $$(x_i)_{\text {min}}$$ and $$(x_i)_{\text {max}}$$ represent the minimum and maximum gray levels in the current set for which $$h((x_i)_{\text {min}})\ne 0$$ and $$h((x_i)_{\text {max}})\ne 0$$, respectively. Basically, the crossover point $$b$$ is the mean gray level value of the interval $$[x_p,x_q]$$. With the function parameters computed in this way, the $$S$$-type membership function adjusts its shape as a function of the set elements.

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May 25, 2017 | Posted by in GENERAL & FAMILY MEDICINE | Comments Off on Fuzzy Measures and Weighted Co-Occurrence Matrix for Segmentation of Brain MR Images

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