Wavelet transform modulus maxima method: Difference between revisions

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{{Short description|methodMethod for detecting a signal's fractal dimension}}

The '''wavelet transform modulus maxima (WTMM)''' is a method for detecting the [[fractal dimension]] of a signal.

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The WTMM was developed out of the larger field of continuous wavelet transforms, which arose in the 1980s, and its contemporary fractal dimension methods.

At its essence, it is a combination of fractal dimension "[[box counting"]] methods and continuous wavelet transforms, where wavelets at various scales are used instead of boxes.

WTMM was originally developed by Mallat and Hwang in 1992 and used for image processing. [httphttps://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=119727&isnumber=3425]. {{Disputed|Original work by Muzy, Bacry, and Arneodo [http://prl.aps.org/abstract/PRL/v67/i25/p3515_1] pre-dates Mallat's and Hwang's|date=January 2023}}

Bacry, Muzy, and Arneodo were early users of this methodology. [http://prl.aps.org/abstract/PRL/v67/i25/p3515_1][http://pre.aps.org/abstract/PRE/v47/i2/p875_1] It has subsequently been used in fields related to signal processing.

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* Alain Arneodo et al. (2008), [[Scholarpedia]], 3(3):4103. [http://www.scholarpedia.org/article/Wavelet-based_multifractal_analysis]

* ''A Wavelet Tour of Signal Processing'', by Stéphane Mallat; {{isbn|012466606X}}; Academic Press, 1999 [http://www.ceremade.dauphine.fr/~peyre/wavelet-tour/] {{Webarchive|url=https://web.archive.org/web/20110228132542/http://www.ceremade.dauphine.fr/~peyre/wavelet-tour/ |date=2011-02-28 }}

* Mallat, S.; Hwang, W.L.;, "Singularity detection and processing with wavelets," ''IEEE Transactions on Information Theory'', volume 38, number 2, pages 617–643, Mar 1992 {{doi|10.1109/18.119727}} [httphttps://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=119727&isnumber=3425]

* Arneodo on Wavelets [http://www.iscpif.fr/tiki-index.php?page=CSSS'08+Arneodo&highlight=towards]{{Dead link|date=August 2025 |bot=InternetArchiveBot |fix-attempted=yes }}

* {{cite journal | last=Muzy | first=J. F. | last2=Bacry | first2=E. | last3=Arneodo | first3=A. | title=Wavelets and multifractal formalism for singular signals: Application to turbulence data | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=67 | issue=25 | date=1991-12-16 | issn=0031-9007 | doi=10.1103/physrevlett.67.3515 | pmid=10044755 | bibcode=1991PhRvL..67.3515M | pages=3515–3518}}

* {{cite journal | last=Muzy | first=J. F. | last2=Bacry | first2=E. | last3=Arneodo | first3=A. | title=Multifractal formalism for fractal signals: The structure-function approach versus the wavelet-transform modulus-maxima method | journal=Physical Review E | publisher=American Physical Society (APS) | volume=47 | issue=2 | date=1993-02-01 | issn=1063-651X | doi=10.1103/physreve.47.875 | pmid=9960082 | bibcode=1993PhRvE..47..875M | pages=875–884| url=https://hal.archives-ouvertes.fr/hal-01557138/file/MBA.pdf }}