TY - GEN

T1 - Fractal dimension estimation using the fast continuous wavelet transform

AU - Vrhel, Michael J.

AU - Lee, Chulhee

AU - Unser, Michael A.

PY - 1995

Y1 - 1995

N2 - 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.

AB - 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.

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M3 - Conference contribution

AN - SCOPUS:0029515091

SN - 0819419281

SN - 9780819419286

T3 - Proceedings of SPIE - The International Society for Optical Engineering

SP - 478

EP - 488

BT - Proceedings of SPIE - The International Society for Optical Engineering

A2 - Laine, Andrew F.

A2 - Unser, Michael A.

A2 - Wickerhauser, Mladen V.

T2 - Wavelet Applications in Signal and Image Processing III. Part 1 (of 2)

Y2 - 12 July 1995 through 14 July 1995

ER -