Spatial and Spectral Descriptions of Stationary Gaussian Fractals
R. A. Rensink, Laboratory for Computational Vision, University of British Columbia, Vancouver, Canada.

UBC CS Technical Report 88-14 (July 1988).

Abstract

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 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, as well as a form of ``locally" self-similar behavior. 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.

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