Localized measurement of image fractal dimension using gabor filters

Abstract

We present a method for making accurate, optimally localized measurements of the fractal dimension of images modeled as locally fractal Brownian surfaces. Fractal Brownian surfaces are good models for the multiscale and irregular image textures arising from natural scenes and for elevation maps of terrain. 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 spectral 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 at every point in the image. We also introduce a variational technique for enforcing a smoothness constraint on the computed fractal dimension function. This technique is implemented using an iterative relaxation algorithm. We demonstrate the method on synthetic images, real images of natural textures, and U.S. Geo-Data digital elevation maps of real terrain. We discuss the ways in which real images depart from the fractal Brownian surface model.