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Complicated self-similarity of terrain surface

Fractal describes the self-similar phenomenon of signal and self-similarity is the most important character of fractal. Pentland provides an excellent explanation of the ruggedness of natural surface. Fractal-based description of image texture has been used effectively in characterization and segmentation of natural scene. A real surface is self-similar over some range of scales, rather than over all scales. That imply self-similarity of a terrain surface is not always so perfect that keep invariable in whole scale space. To describe such self-similarity distribution, a self-similarity curve could be plotted and was divided into several linear regions. We present a new parameter called Self-similarity Degree (SD) in the similitude of information entropy to denote such self-similarity distribution. In addition, one general characterization of self-similarities is result of physical processes. Terrain surface are created by the interactional inogenic and exogenic processes. Hereby, we introduce self-similarity analysis and multifractal singularity spectrum to describe such complex physical field. By the self-similarity analysis and singularity spectrum, the different self-similar structures and the interaction of processes in terrain surface were depicted. Our studies have shown that self-similarity is a relative notion and natural scenes own abundant self-similar structures. Moreover, noises always destroy the self-similarity of original natural surface and change the singularity distribution of original surface.