103.

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and a box, whose Euclidean dimensions are equal to one and two, respectively.

The coastline measurement problem demonstrates another difficulty: the apparent length of the coastline changes as the step size of the measuring tool increases or decreases, i.e. it is not consistent. Moreover, whatever the step length l of measuring instrument selected, coastline features whose size is smaller than l will not be measured. It is clear that the apparent coastline length depends not only on the geometry of coastline itself, but also relates to the size of measuring tool.

Mandelbrot was attracted by this measurement inconsistency problem. He found that the measurement inconsistency is due to the loss of fractional features (features whose size is smaller than the step size of the measuring instrument). In order to contain the contribution of these fractional features, Mandelbrot pointed out that we must generalise the notion of dimension to include these fractional dimensions. Once these ‘lost’ fractional features are compensated for, the fractal dimension will yield a uniform measure at any measurement scale.

5.1.2 Estimation of the fractal dimension

There are several methods of estimating fractal dimension. A detailed demonstration of fractal dimension estimation via the wavelet transform is provided by Mallat (1989). He pointed out that if the signal is truly fractal, then the fractal dimension will show some relation to the power spectrum and thus can be estimated. Peleg et al. (1984) use the -blanket method to obtain an estimate of fractal dimension, which is a generalised version due to Mandelbrot (1982). Pentland (1984) fitted a model called fractional Brownian motion (FBM) to the image in order to measure its fractal dimension. The box-counting method is another way to obtain an estimate of the fractal dimension. Box-counting approaches are presented by Clarke (1986), Gonzato (1998), Keller and Chen (1989), Sarkar and Chaudhuri (1994), and Voss (1986). Since the methods used in the FBM and box-counting approaches are straightforward and efficient, they are discussed in detail in the following sections.

5.1.2.1 Fractal Brownian motion (FBM)

Pentland (1984) presents a method to estimate the fractal dimension of an object by fitting a FBM model to the frequency domain representation of the image intensity surface. A random function f(t) is a fractional Brownian function if the following relationship holds for all t and Δt:

(5.2)

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