120.

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Page 205

(5.26)

where l is the sub-box length, as specified previously. We can thus use D(α) as the fractal dimension associated with the sub-box linked by a. Note that D(α) is also called the alpha spectrum, i.e. the spectrum of the subsets’ fractal dimensions, which is different from the global set. The function f(α) has been demonstrated to be a Legendre transform of the function D(q) (Parisi and Frish, 1985). Both D(α) and D(q) display the following relationship:

(5.27)

The function D(q) is called the q moment generalised fractal dimension and is shown by Hentchel and Procaccia (1983) to be:

(5.28)

where u_i denotes the percentage of points contained in sub-box i, and q is a real number. In order to obtain a continuous function of D(q), the case for q=1 is dealt with separately, as follows:

(5.29)

Tso (1997) provides details of this derivation. A continuous function D(q) has therefore been specified. The function D(q), which is derived based on the concept of the multifractal dimension, provides a convenient way to characterise a local set’s property by introducing different values of the weighting factor q.

5.1.4 Estimation of q moment generalised fractal dimension

Using the definition of function D(q) it is possible to extend the method of fractal Brownian motion (FBM) and box counting to estimate D(q), for use as a basis for image segmentation. However, different approaches may generate different values of D(q), resulting in variations in image segmentation results. It is therefore of interest to carry out a comparison of the competing methods.

The FBM method (Equation (5.4)) and three box-counting methods,

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