Parameter estimation and denoising of 2-D noisy fractional Brownian motion using non-orthogonal wavelets

Abstract

Fractional Brownian motion (fBm) is a non-stationary stochastic model, which has a 1/f spectrum and statistical self-similar property. We extend the proposed methods of Hwang to an isotropic 2-D noisy fBm image. The extension is not straightforward; although one can obtain the fractal parameter of an isotropic fBm by averaging of the estimated fractal parameters from several directions by means of the 1-D fractal parameter estimation algorithm, this approach does not perform well in practice. It was shown by Hwang that it requires more than 1000 sampled points for a robust 1-D fractal parameter estimation. For a median size image (say with size 256 by 256 or smaller), there is not enough pixels at each direction for a robust 1-D fractal parameter estimation. Thus, alternative methods must be developed in order that the robustness fractal estimation from a noisy fBm image with small size can be achieved. In this paper, we show that the wavelet transform of an isotropic fBm image at each scale is a two-dimensional weakly stationary process at both the horizontal and vertical directions. Thus, robust fractal parameter estimation can be obtained from two-dimensional wavelet coefficients, even for a small noisy fBm image. We propose a fractal parameter estimation algorithm which formulates the robust fractal parameter estimation problem as the characterization of a composite singularity from the autocorrelation of wavelet transforms of a noisy fBm image.