104.

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F(y) is a cumulative distribution function with the self-affine property, which indicates that Equation (5.2) is still valid given affine transform of the function f(t), and H is called the Hurst coefficient, which determines the distribution of F(y). It should be noted that in Equation (5.2), t and Δt can be represented by vector quantities. Hence, this formula can be extended to higher Euclidean dimensions. The term denotes the length of vector Δt. In the case of an image, is the distance between two pixels of interest. The relationship between fractal dimension D and H is given by:

(5.3)

Equation (5.2) implies that, for every Δt,

(5.4)

E(|Δf_Δt|) is the expected value of the change in the intensity surface over Δt. C is a constant, equal to the mean of the random variable |y| (Yokoya et al., 1989). One can therefore measure the quantities of E(|Δf_Δt|) for various Δt, then use a least-square fit in the log-log domain to obtain H (see Equation (5.4)) and thus determine the fractal dimension D. Based on the above description it is clear that for the same distribution function F(y), a higher (or lower) value of H will result in lower (higher) value of dimension D, which indicates a relatively smooth (rough) image intensity surface. D is seen to be directly related to image texture.

5.1.2.2 Box-counting methods

Box-counting methods are based on Mandelbrot’s self-similarity equation. Mandelbrot (1977, 1982) pointed out that one criterion of a surface being fractal is based on the self-similarity property. Consider a pattern A in two-dimensional space. The pattern A is defined to be self-similar if A is the union of N distinct (non-overlapping) copies of itself, each of which is similar to A scaled down by a ratio r in each dimension. Thus, the fractal dimension D of A can be expressed by the following equations:

(5.5)

Using this equation, the fractal dimensions of the patterns shown in Figure 5.1 can be computed. In Figure 5.1a, the scale r on each dimension is ½, the number of copies N is 3, the fractal dimension D can be obtained as

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