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UBC CS TR-88-14 Summary

Spatial and Spectral Description of Stationary Gaussian Fractals, July 1988 R. Rensink

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. \n 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.


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