Figure 2.4: Accumulation local moments response. (a) Original image. (b) Accumulation local moment of order (3,1).

throughout the region of interest. The parameter a controls the shape of the logistic function. Therefore each textural feature will be the result of the application of the nonlinear operator to the computed moments. If n = k ■ l moments are computed over the image, then the dimension of the feature vector will be n. Hence, a n-dimensional point is associated with each pixel of the image.

Figure 2.4 shows the response of moment (3,1) on an IVUS image. In this figure, the response seems to have a smoothing and enhancing effect, clearly resembling diffusion techniques. Fractal Analysis

Another classic tool for texture description is the fractal analysis [13, 33], characterized by the fractal dimension. We talk roughly about fractal structures when a geometric shape can be subdivided in parts, each of which are approximately a reduced copy of the whole (this property is also referred as self-similarity). The introduction of fractals by Mandelbrot [27] allowed a characterization of complex structures that could not be described by a single measure using Euclidean geometry. This measure is the fractal dimension, which is related to the degree of irregularity of the surface texture.

The fractal structures can be divided into two subclasses: the deterministic fractals and the random fractals. Deterministic fractals are strictly self-similar, that is, they appear identical over a range of magnification scales. On the other hand, random fractals are statistical self-similar. The similarity between two scales of the fractal is ruled by a statistical relationship.

The fractal dimension represents the disorder of an object. The higher the dimension, the more complex the object is. Contrary to the Euclidian dimension, the fractal dimension is not constrained to integer dimensions.

The concept of fractals can be easily extrapolated to image analysis if we consider the image as a three-dimensional surface in which the height at each point is given by the gray value of the pixel.

Different approaches have been proposed to compute the fractal dimension of an object. Herein we consider only three classical approaches: box-counting, Brownian motion, and Fourier analysis.

Box-Counting. The box-counting method is an approximation to the fractal dimension as it is conceptually related to self-similarity.

In this method the object to be evaluated is placed on a square mesh of various sizes, r. The number of mesh boxes, N, that contain any part of the fractal structure are counted.

It has been proved that in a self-similar structures there is a relationship between the reduction factor r and the number of divisions N into which the structure can be divided:

where D is the self-similarity dimension. Therefore, the fractal dimension can be easily written as

This process is done at various scales by altering the square size r. Therefore, the box-counting dimension is the slope of the regression line that better approximates the data on the plot produced by log N x log 1 /r.

Fractal Dimension from Brownian Motion. The fractal dimension is found by considering the absolute intensity difference of pixel pairs, I (pi) — I(p2), at different scales. It can be shown that for a fractal Brownian surface the following relationship must be satisfied:

where E is the mean and H the Hurst coefficient. The fractal dimension is related to H in the following way: D = 3 — H .In the same way than the former

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