Spatial and Spectral Descriptions of Stationary Gaussian Fractals

A general treatment of stationary Gaussian fractals is presented. Relations are established between the fractal properties of an $n$-dimensional random field and the form of its correlation function and power spectrum. These relations are used to show that the second-order parameter $H$ commonly used to describe fractal texture (e.g., in [4][5]) is insufficient to characterize all fractal aspects of the field. A larger set of measures --- based on the power spectrum --- is shown to provide a more complete description of fractal texture.

Several interesting types of "non-fractal" self-similarity are also developed. These include a generalization of the fractional Gaussian noises of Mandelbrot and van Ness [6], as well as a form of "locally" self-similar behaviour. It is shown that these have close relations to the Gaussian fractals, and consequently, that textures containing these types of self-similarity can be described by the same set of measures as used for fractal texture.