On Synthesizing Discrete Fractional Brownian Motion with Applications to Image Processing

Abstract

This paper evaluates the following four methods for synthesizing discrete fractional Brownian motion (dfBm): sampled fBm, displaced interpolation, spectral synthesis, and Karhunen–Loeve-like wavelet expansion with respect to the following questions: Does the candidate dfBm have the correct second-order statistics? Does the candidate dfBm have the correct fractal dimension? To estimate the fractal dimension we apply spectral linear regression, multiresolution energy analysis, and maximum likelihood estimation. Running an estimation routine on synthesized data raises the final question: How do the assumptions of the estimation method interact with the assumptions of the synthesis method to produce the results of the various trials? Our main conclusions relative to these questions are: (1) Sampled fBm is the only one of the four methods which has the correct second-order statistics, but only spectral synthesis produces a stationary signal. (2) The fractal dimension of signals produced by sampled fBm, displaced interpolation, and spectral synthesis can be estimated with reasonable accuracy. On the other hand, the wavelet signals are only slightly sensitive to the input values ofH. (3) Analysis routines which make assumptions not made by the synthesis routine generally yield poor estimates ofH, whereas a match of assumptions generally produces good estimates.