Two-dimensional fractional Brownian motion: wavelet analysis and synthesis

Two-dimensional fractional Brownian motion (2D FBM) is a non-stationary random process that displays fractal (or self-similar properties). Its correlation function and power spectral density both follow power-law forms. Its fractal nature is characterized by a self-similarity parameter, termed the Hurst (1951) exponent H. We first demonstrate how a power-law spectral density arises from the definition of 2D FBM. We then consider the wavelet transform of 2D FBM, and show how it can be used to estimate H. Finally, we consider the converse problem and demonstrate how wavelets can be used to synthesize processes with power spectral densities close to those of 2D FBM. The results demonstrate that from an analysis point of view, the wavelet transform of 2D FBM retains the self-similar properties of the original signal, and can be used to estimate the self-similarity parameter of the process. This parameter is important in texture recognition, and as a measure of the roughness of a fractal surface