### 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.

Original language | English |
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Title of host publication | Proceedings of SPIE - The International Society for Optical Engineering |

Editors | Andrew F. Laine, Michael A. Unser, Mladen V. Wickerhauser |

Pages | 478-488 |

Number of pages | 11 |

Volume | 2569 |

Edition | 2/- |

Publication status | Published - 1995 Dec 1 |

Event | Wavelet Applications in Signal and Image Processing III. Part 1 (of 2) - San Diego, CA, USA Duration: 1995 Jul 12 → 1995 Jul 14 |

### Other

Other | Wavelet Applications in Signal and Image Processing III. Part 1 (of 2) |
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City | San Diego, CA, USA |

Period | 95/7/12 → 95/7/14 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Electrical and Electronic Engineering
- Condensed Matter Physics

### Cite this

*Proceedings of SPIE - The International Society for Optical Engineering*(2/- ed., Vol. 2569 , pp. 478-488)

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*Proceedings of SPIE - The International Society for Optical Engineering.*2/- edn, vol. 2569 , pp. 478-488, Wavelet Applications in Signal and Image Processing III. Part 1 (of 2), San Diego, CA, USA, 95/7/12.

**Fractal dimension estimation using the fast continuous wavelet transform.** / Vrhel, Michael J.; Lee, Chulhee; Unser, Michael A.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Fractal dimension estimation using the fast continuous wavelet transform

AU - Vrhel, Michael J.

AU - Lee, Chulhee

AU - Unser, Michael A.

PY - 1995/12/1

Y1 - 1995/12/1

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.

UR - http://www.scopus.com/inward/record.url?scp=0029515091&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0029515091&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:0029515091

SN - 0819419281

SN - 9780819419286

VL - 2569

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.

ER -