The Variation Method: A Technique To Estimate The Fractal Dimension Of Surfaces

B. Dubuc; C. Roques-Carmes; C. Tricot; S. W. Zucker

doi:10.1117/12.976511

13 October 1987 The Variation Method: A Technique To Estimate The Fractal Dimension Of Surfaces

B. Dubuc, C. Roques-Carmes, C. Tricot, S. W. Zucker

Proceedings Volume 0845, Visual Communications and Image Processing II; (1987) https://doi.org/10.1117/12.976511
Event: Cambridge Symposium on Optics in Medicine and Visual Image Processing, 1987, San Diego, CA, United States

Abstract

There are many definitions of the fractal dimension of an object, including box dimension. Bouligand-Minkowski dimension, and intersection dimension. Although they are all equivalent in the continuous domain, when discretized and applied to digitized data, they differ substantially. We show that the standard implementations of these definitions on self-afline curves with known fractal dimension (Weierstrass-Mandelbrot, Kiesswetter, fractional Brownian motion) yield results with errors ranging up to 5 or 10%. An analysis of the source of these errors led to a new algorithm in 1-D. called the variation method, which yielded accurate results. The variation method uses the notion of e-variation to measure the amplitude of the one-dimensional function in an e-neighborhood. The order of growth of the integral of the e-variation, as E tends toward zero, is directly related to the fractal dimension. In this paper, we extend the variation method to higher dimensions and show that, in the limit, it is equivalent to the classical box counting method. The result is an algorithm for reliably estimating the fractal dimension of surfaces or, more generally, graphs of functions of several variables. The algorithm is tested on surfaces with known fractal dimension and is applied to the study of rough surfaces.

Citation Download Citation

B. Dubuc, C. Roques-Carmes, C. Tricot, and S. W. Zucker "The Variation Method: A Technique To Estimate The Fractal Dimension Of Surfaces", Proc. SPIE 0845, Visual Communications and Image Processing II, (13 October 1987); https://doi.org/10.1117/12.976511

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