Fractal dimension estimation using the fast continuous wavelet transform

Michael J. Vrhel; Chulhee Lee; Michael A. Unser

doi:10.1117/12.217603

1 September 1995 Fractal dimension estimation using the fast continuous wavelet transform

Michael J. Vrhel, Chulhee Lee, Michael A. Unser

Proceedings Volume 2569, Wavelet Applications in Signal and Image Processing III; (1995) https://doi.org/10.1117/12.217603
Event: SPIE's 1995 International Symposium on Optical Science, Engineering, and Instrumentation, 1995, San Diego, CA, United States

Abstract

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.

Citation Download Citation

Michael J. Vrhel, Chulhee Lee, and Michael A. Unser "Fractal dimension estimation using the fast continuous wavelet transform", Proc. SPIE 2569, Wavelet Applications in Signal and Image Processing III, (1 September 1995); https://doi.org/10.1117/12.217603

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