<title>Optimally localized estimation of the fractal dimension</title>

Abstract

We present a method for making accurate measurements of the instantaneous fractal dimension of (1) images modeled as fractal Brownian surfaces, and (2) images of physical surfaces modeled as fractal Brownian surfaces. Fractal Brownian surfaces have the property that their apparent roughness increases as the viewing distance decreases. Since this true of many natural surfaces, fractal Brownian surfaces are excellent candithtes for modeling rough surfaces. To obtain accurate local values of the fractal dimension, spatio-spectrally localized measurements are necessary. Our method employs Gabor filters, which optimize the conflicting goals of spatial and speciral localization as constrained by the functional uncertainty principle. The outputs from multiple Gabor filters are fitted to a fractal power-law curve whose parameters determine the fractal dimension. The algorithm produces a local value of the fractal dimension for every point in the image. We also introduce a variational technique for producing a fractal dimension function which varies smoothly across the image. This technique is implemented using an iterative relaxation algorithm. A test of the method on 50 synthetic images of known global fractal dimensions shows that the method is accurate with an error of approximately 4.5% for fractal Brownian images and approximately 8.5% for images of physical fractal Brownian surfaces.