Classifying Mammography Images by Using Fuzzy Cognitive Maps and a New Segmentation Algorithm

Abstract

Mammography is one of the best techniques for the early detection of breast cancer. In this chapter, a method based on fuzzy cognitive map (FCM) and its evolutionary-based learning capabilities is presented for classifying mammography images. The main contribution of this work is two-fold: (a) to propose a new segmentation approach called the threshold based region growing (TBRG) algorithm for segmentation of mammography images, and (b) to implement FCM method in the context of mammography image classification by developing a new FCM learning algorithm efficient for tumor classification. By applying the proposed (TBRG) algorithm, a possible tumor is delineated against the background tissue. We extracted 36 features from the tissue, describing the texture and the boundary of the segmented region. Due to the curse of dimensionality of features space, the features were selected with the help of the continuous particle swarm optimization algorithm. The FCM was trained using a new evolutionary approach based on the area under curve (AUC) of the output concept. In order to evaluate the efficacy of the presented scheme, comparisons with benchmark machine learning algorithms were conducted and known metrics like ROC, AUC were calculated. The AUC obtained for the test data set is 87.11%, which indicates the excellent performance of the proposed FCM.

Appendix

In this appendix, the features extracted from the segmented images are described. A total of 36 features that describe the texture and the boundaries of the segmented region were extracted. The designations of these features are given in Table 5, followed by a brief explanation regarding each extracted feature.

Table 5 Names of the extracted features

• 1. Circularity

The circularity of a region, which shows its resemblance to a circle, is obtained from Eq. (7)

$$ Circularity = \frac{{P^{2} }}{A} $$

(7)

Here, P and A denote the circumference and the area of a segmented region, respectively.

• 2. Area

The area of a segmented region is equal to the total number of pixels which are members of this region.

• 3. Features related to radial length

In order to determine this feature, it is necessary to first obtain the center of the segmented region by Eq. (8). Then the Euclidean distance of each pixel on the contour (Boundary of the segmented region) is calculated from the geometrical center. The average and the standard deviation of these distances constitute the two features of this section.

$$ X = \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} x_{i} .\quad Y = \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} y_{i} $$

(8)

Here, N is the number of pixels on the contour, x_i and y_i are the x,y coordinates of the ith pixel on the contour, and X and Y are the x, y coordinates of the segmented region’s geometrical center.

• 4. Entropy of the segmented region

The entropy of the segmented region’s brightness intensity is obtained from Eq. (9). The entropy is a measure of the randomness of a random variable.

$$ E = - \mathop \sum \limits_{i = 1}^{N} p_{i} { \log }\left( {p_{i} } \right) $$

(9)

where, N denotes the number of brightness levels and p_i is the probability of having a pixel with brightness level i in the segmented region.

• 5. Features related to fractal index

By measuring the variations of details with respect to scale, fractal dimension can provide a criterion for the complexity of a segmented region [39]. The fractal dimension of a segmented region can be computed by means of Eq. (10).

$$ N = N_{0} R^{ - D} $$

(10)

Here, N denotes the number of boxes superimposed to the segmented area, N₀ is an arbitrary constant, R is the size of different boxes and D is the fractal dimension. N and R can be obtained by using the box counting method [40]. In view of Eq. (10), if the log^N-log^R diagram is modeled on a line, the slope (i.e. the fractal dimension) and the intercept of this line can be used as two features. Also, the dispersion variance of the slopes of lines in the log^N-log^R diagram could be considered as another feature.

• 6. Eccentricity

Eccentricity shows the degree of lengthening of the segmented region [41], and it is obtained from the eigenvalues of Matrix A whose entries can be defined with regards to Eq. (11).

$$ A_{11} = \mathop \sum \limits_{i = 1}^{N} \left( {x_{i} - X} \right)^{2} \cdot A_{22} = \mathop \sum \limits_{i = 1}^{N} \left( {y_{i} - Y} \right)^{2} \cdot { }A_{12} = A_{21} = \mathop \sum \limits_{i = 1}^{N} \left( {x_{i} - X} \right)\left( {y_{i} - Y} \right) $$

(11)

In the above equation, A_ij is the entry related to the ith row and jth column of Matrix A, N is the number of pixels on the boundary, x_i and y_i represent the coordinates of the ith pixel on the boundary, and X and Y denote the coordinates of the segmented region’s geometrical center.

• 7. Second invariant moment

The second and the third moments of an image can be defined so that they are robust against variables like rotation and scale. A number of these moments have been introduced in [42]. We have extracted the second invariant moment as a feature for the segmented region.

• 8. Entropy of contour gradient

For extracting this feature, by using the Sobel operator, the gradient direction of the pixels on the segmented region’s boundary is determined. Then, by considering this gradient direction and Fig. 4, the histogram of gradient direction is obtained for the 8-neighbor case, and by 8 bins.

Fig. 4

The bins of gradient direction histogram for the case of 8 neighbors (each bin covers a total of 45º)

After establishing the gradient direction histogram, the entropy of contour gradient can be obtained from Eq. (12).

$$ E = - \mathop \sum \limits_{i = 1}^{8} p_{i} { \log }\left( {p_{i} } \right) $$

(12)

where, p_i is the probability of finding an arbitrary pixel on the boundary with maximum fluctuations in the direction of histogram’s ith bin.

• 9. Average brightness intensity

The mean brightness intensity of the segmented region is extracted as a feature.

• 10. Standard deviation of brightness intensity

The standard deviation of the brightness intensities of the pixels within the segmented region is extracted as a feature.

• 11. Features related to GLCM

The grey level co-occurrence matrix (GLCM) has been extensively used to describe the image textures. In order to extract the features associated with GLCM, the tumor tissue texture is extracted by using the bounding box method, and then the GLCM matrix is determined for this region. Subsequently, 14 features are extracted, which are definable from this matrix.

• 12. Histogram of gradients

In this approach, which is the simplified method of the histogram of oriented gradients [43], the whole texture of the segmented region is extracted first by using the bounding box method, and the extracted image is considered as a cell. For this cell, a 9-bin histogram is obtained by considering the size and gradient direction of the existing pixels and these bins are used as features.