4.2.1 Fourier dimension and fractal dimension

As with many other techniques of digital signal and image processing, computation of the fractal dimension can be undertaken in real space (processing the data directly) or in transform space (processing the data after taking an appropriate integral transform). In the latter case, use can be made of the Fourier transform, as the PSDF of a fractal signal or image has the expected form. This important relationship between a random scaling fractal and its PSDF is introduced later, where it forms the basis for a more general discussion on stochastic modelling using PSDF models.

In general, there is no unique and general rule for computing the fractal dimension. A large number of algorithms have been developed over the past 15 years for computation of the fractal dimension. These can be broadly categorized into two families.

So far, in this book, models for an RSF field have been expressed in terms of the Fourier dimension q alone. In addition, the appropriate ranges that should be attributed to the Fourier dimension for a fractal signal and for a fractal or Mandelbrot surface have been left ambiguous. In this section, the relationship between the fractal dimension D and the Fourier dimension q is quantified. The basic result is derived in [17], where it is shown that the Fourier dimension q and the box-counting dimension D (the fractal dimension) for a fractal signal are related as follows:

In addition, we note that the relationship...