Hx y exp I

Figure 2.11: The filter set in the spatial-frequency domain.

where u and $ are the frequency and phase of the sinusoidal plane wave along the x axis and ax and ay are the space constants of the Gaussian envelope along the x and y axis, respectively. Filters at different orientations can be created by rigid rotation of x-y coordinate system.

An interesting property of this kind of filters is their frequency and orientation-selection. This fact is better displayed in the frequency domain. Figure 2.11 shows the filter area in the frequency domain. We can observe that each of the filters has a certain domain defined by each of the leaves of the Gabor "rose." Thus, each filter responds to a certain orientation and at a certain detail level. Wider the range of orientations, smaller the space filter dimensions and smaller the details captured by the filter, as bandwidth in the frequency domain is inversely related to filter scope in the space domain. Therefore, Gabor filters provide a trade-off between localization or resolution in both the spatial and the spatial-frequential domains. As it has been mentioned, different filters emerge from rotating the x-y coordinate system. For practical approaches one can use four angles 00 = 0°, 45°, 90°, 135°. For an image array of N pixels (with N power of 2), the following values of uo are suggested [25,37]:

cycles per image width. Therefore, the orientations and bandwidth of such filters vary with 45° and 1 octave. These parameters are chosen because there is physiologic evidences of frequency bandwidth of simple cells in visual cortex being of about 1 octave, and Gabor filters try to mimic part of the human perceptual system.

The Gabor function is an approximation to a wavelet. However, though admissible, it does not result in an orthogonal decomposition, and therefore, a transformation based on Gabor's filters is redundant. On the other hand, Gabor filtering is designed to be nearly orthogonal, reducing the amount of overlap between filters.

Figure 2.12 shows different responses for different filters of the spectrum. Figures 2.12(a) and 2.12(b) correspond to the inner filters with reduced frequency bandwidth displayed in Fig. 2.11. It can be seen that they deliver only

Figure 2.12: Gabor filter bank example responses. (a) Gabor vertical energy of a coarse filter response. (b) Gabor horizontal energy of a coarse filter response. (c) Gabor vertical energy of a detail filter response. (d) Gabor horizontal energy of a detail filter response.

Figure 2.12: Gabor filter bank example responses. (a) Gabor vertical energy of a coarse filter response. (b) Gabor horizontal energy of a coarse filter response. (c) Gabor vertical energy of a detail filter response. (d) Gabor horizontal energy of a detail filter response.

Table 2.1: Dimensionality of the feature space provided by the texture feature extraction process

Method

Space dimension

Co-occurrence matrix measures Accumulation local moments Fractal Dimension Local Binary Patterns Derivative of Gaussian Wavelets Gabor's filters

48 81 1

3 60 31 20

coarse information of the structure and the borders are far from the original location. In the same way, Figs. 12(c) and 12(d) are filters located on a further ring, and therefore respond to details in the image.

It can be observed that the feature extraction process is a transformation of the original two-dimensional image domain to a feature space that probably will have different dimensions. In some cases, the feature space remains low, as in fractal dimension and local binary patterns, that with very few features try to describe the texture present in the image. However, several feature spaces require higher dimensions, such as accumulation local moments, co-occurrence matrix measures, or derivatives of Gaussian. Table 2.1 shows the dimensionality of the different spaces generated by the feature extraction process in our texture-based IVUS analysis.

The next step after the feature extraction is the classification process. As a result of the disparity of the dimensionality of the feature spaces, we have to choose a classification scheme that is able to deal with high dimensionality feature data.

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