## Feature Spaces

Gray-level thresholding is not enough for robust tissue characterization. Therefore, it is generally approached as a texture discrimination problem. This line of work is a classical extension of previous works on biological characterization, which also relies on texture features as has been mentioned in the former section. The co-occurrence matrix is the most favored and well known of the texture feature extraction methods due to its discriminative power in this particular problem but it is not the only one nor the fastest method available. In this section, we make a review of different texture methods that can be applied to the problem in particular, from the co-occurrence matrix measures method to the most recent texture feature extractor, local binary patterns.

To illustrate the texture feature extraction process we have selected a set of techniques basing our criterion of selection on the most widespread methods for tissue characterization and the most discriminative feature extractors reported in the literature [21].

Basically, the different methods of feature extraction emphasize on different fundamental properties of the texture such as scale, statistics, or structure. In this way, under the nonelemental statistics property we can find two well-known techniques, co-occurrence methods [22] and higher order statistics represented by moments [23]. Under the label of scale property we should mention methods such as derivatives of Gaussian [24], Gabor filters [25], or wavelet techniques [26]. Regarding structure-related measures there are methods such as fractal dimension [27] and local binary patterns [28].

To introduce the texture feature extraction methods we divide them into two groups: The first group, that forms the statistic-related methods, is comprised of co-occurrence matrix measures, accumulation local moments, fractal dimension, and local binary patterns. All these methods are somehow related to statistics. Co-occurrence matrix measures are second-order measures associated to the probability density function estimation provided by the co-occurrence matrix. Accumulation local moments are directly related to statistics. Fractal dimension is an approximation of the roughness of a texture. Local binary patterns provides a measure of the local inhomogeneity based on an "averaging" process. The second group, that forms the analytic kernel-based extraction techniques, comprises Gabor bank of filters, derivatives of Gaussian filters, and wavelet decomposition. The last three methods are derived from analytic functions and sampled to form a set of filters, each focused on the extraction of a certain feature.

2.2.1 Statistic-Related Methods 2.2.1.1 Co-occurrence Matrix Approach

In 1962 Julesz [29] showed the importance of texture segregation using second-order statistics. Since then, different tools have been used to exploit this issue. The gray-level co-occurrence matrix is a well-known statistical tool for extracting second-order texture information from images [22]. In the co-occurrence method, the relative frequencies of gray-level pairs of pixels at certain relative displacement are computed and sorted in a matrix, the co-occurrence matrix P. The co-occurrence matrix can be thought of as an estimate of the joint probability density function of gray-level pairs in an image. For G gray levels in the image, P will be of size G x G. If G is large, the number of pixel pairs contributing to each element, p^ j in P is low, and the statistical significance poor. On the other hand, if the number of gray levels is low, much of the texture information may be lost in the image quantization. The element values in the matrix, when normalized, are bounded by [0, 1], and the sum of all element values is equal to 1.

P(i, j, D,G) = P(I(l, m) = i and I(l + D cos(0), m + D sin(0)) = j where I(l, m) is the image at pixel (l, m), D is the distance between pixels,

+1 | ||||||

Figure 2.2: Co-occurrence matrix explanation diagram (see text).

and 0 is the angle. It has been proved by other researchers [21, 30] that the nearest neighbor pairs at distance D at orientations 0 = {0°, 45°, 90°, 135°} are the minimum set needed to describe the texture second-order statistic measures. Figure 2.2 illustrates the method providing a graphical explanation. The main idea is to create a "histogram" of the occurrences of having two pixels of certain gray levels at a determined distance with a fixed angle. Practically, we add one to the cell of the matrix pointed by the gray levels of two pixels (one pixel gray level gives the file and the other the column of the matrix) that fulfill the requirement of being at a certain predefined distance and angle.

Once the matrix is computed several characterizing measures are extracted. Many of these features are derived by weighting each of the matrix element values and then summing these weighted values to form the feature value. The weight applied to each element is based on a feature-weighing function, so by varying this function, different texture information can be extracted from the matrix. We present here some of the most important measures that characterize the co-occurrence matrices: energy, entropy, inverse difference moment, shade, inertia, and promenance [30]. Let us introduce some notation for the definition of the features:

P(i, j) is the (i, j)th element of a normalized co-occurrence matrix

With the above notation, the features can be written as follows:

Hence, we create a feature vector for each of the pixels by assigning each feature measure to a component of the feature vector. Given that we have four different orientations and the six measures for each orientation, the feature vector is a 24-dimensional vector for each pixel and for each distance. Since we have used two distances D = 2 and D = 3, the final vector is a 48-dimensional vector.

Figure 2.3 shows responses for different measures on the co-occurrence matrices. Although a straightforward interpretation of the feature extraction response is not easy, some deduction can be made by observing the figures. Figure 2.3(b) shows shade measure; as its name indicates it is related to the shadowed areas in the image, and thus, localizing the shadowing behind the calcium plaque. Figure 2.3(c) shows inverse different moment response, this measure seems to be related to the first derivative of the image, enhancing contours. Figure 2.3(d) depicts the output for the inertia measure, which seems to have some relationship with local homogeneity of the image.

Figure 2.3: Response of an IVUS image to different measures of the cooccurrence matrix. (a) Original image, (b) measure shade response, (c) inverse different moment, and (d) inertia.

Figure 2.3: Response of an IVUS image to different measures of the cooccurrence matrix. (a) Original image, (b) measure shade response, (c) inverse different moment, and (d) inertia.

### 2.2.1.2 Accumulation Local Moments

Geometric moments have been used effectively for texture segmentation in many different application domains [23]. In addition, other kind of moments have been proposed: Zernique moments, Legendre moments, etc. By definition, any set of parameters obtained by projecting an image onto a two-dimensional polynomial basis is called moments. Then, since different sets of polynomials up to the same order define the same subspace, any complete set of moments up to given order can be obtained from any other set of moments up to the same order. The computation of some of these sets of moments leads to very long processing times, so in this section a particular fast computed moment set has been chosen. This set of moments is known as the accumulation local moments. Two kind of accumulation local moments can be computed, direct accumulation and reverse accumulation. Since direct accumulation is more sensitive to round off errors and small perturbations in the input data [31], the reverse accumulation moments are recommendable.

The reverse accumulation moment of order (k — 1, l — 1) of matrix Iab is the value of Iab[1,1] after bottom-up accumulating its column k times (i.e., after applying k times the assignment Iab [a — i, j] <— Iab [a — i, j] + Iab [a — i + 1, j], for i = 0 to a — 1, and for j = 1 to b), and accumulating the resulting first row from right to left l times (i.e., after applying l times the assignment Iab [1, b — j] Iab [1, b — j] + Iab [1, b — j + 1], for j = 1 to b — 1). The reverse accumulation moment matrix is defined so that Rmn[k.l] is the reverse accumulation moment of order (k — 1, l — 1).

Consider the matrix in the following example:

According to the definition, its reverse accumulation moment of order (1,2) requires two column accumulations,

5 |
4 |
6\ | |

5 |
3 |
4 | |

4 |
2 |
3 |
and three right to left accumulations of the first row: Then it is said that the reverse accumulation moment of order (1,2) of the former matrix is 119. The set of moments alone is not sufficient to obtain good texture features in certain images. Some iso-second order texture pairs, which are preattentively discriminable by humans, would have the same average energy over finite regions. However, their distribution would be different for the different textures. One solution suggested by Caelli is to introduce a nonlinear transducer that maps moments to texture features [32]. Several functions have been proposed in the literature: logistic, sigmoidal, power function, or absolute deviation of feature vectors from the mean [23]. The function we have chosen is the hyperbolic tangent function, which is logistic in shape. Using the accumulation moments image Im and a nonlinear operator |tanh(a(Im — Im)\ an "average" is performed |

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