Fractal dimension estimation using the fast continuous wavelet transform

We first review a method for the characterization of fractal signals introduced by Muzy et al. This approach uses the continuous wavelet transform (CWT) and considers how the wavelet values scale along maxima lines. The method requires a fine scale sampling of the signal and standard dyadic algorithms are not applicable. For this reason, a significant amount of computation is spent evaluating the CWT. To improve the efficiency of the fractal estimation, we introduced a general framework for a faster computation of the CWT. The method allows arbitrary sampling along the scale axis, and achieves O(N) complexity per scale where N is the length of the signal. Our approach makes use of a compactly supported scaling function to approximate the analyzing wavelet. We discuss the theory of the fast wavelet algorithm which uses a duality principle and recursive digital filtering for rapid calculation of the CWT. We also provide error bounds on the wavelet approximation and show how to obtain any desired level of accuracy. Finally, we demonstrate the effectiveness of the algorithm by using it in the estimation of the generalized dimensions of a multi-fractal signal.