Wavelet transform modulus maxima method - Revision history

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2024-07-31T05:20:49Z

Move 2 urls. Wayback Medic 2.5 per WP:URLREQ#ieee.org

← Previous revision		Revision as of 05:20, 31 July 2024
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	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.		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 [~~http~~://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}}		WTMM was originally developed by Mallat and Hwang in 1992 and used for image processing [https://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.		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]		* 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/]		* ''A Wavelet Tour of Signal Processing'', by Stéphane Mallat; {{isbn\|012466606X}}; Academic Press, 1999 [http://www.ceremade.dauphine.fr/~peyre/wavelet-tour/]
	* 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}} [~~http~~://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=119727&isnumber=3425]		* 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}} [https://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]		* Arneodo on Wavelets [http://www.iscpif.fr/tiki-index.php?page=CSSS'08+Arneodo&highlight=towards]
	* {{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=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}}

AnomieBOT: Dating maintenance tags: {{Disputed}}

2023-01-09T10:32:20Z

Dating maintenance tags: {{Disputed}}

← Previous revision		Revision as of 10:32, 9 January 2023
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	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.		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 [http://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}}		WTMM was originally developed by Mallat and Hwang in 1992 and used for image processing [http://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.		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.

RydinG: /* History */ Statement regarding original work by Mallat and Hwang from 1992 is disputed given the existence of a prior work by Muzy et al from 1991.

2023-01-09T08:31:19Z

History: Statement regarding original work by Mallat and Hwang from 1992 is disputed given the existence of a prior work by Muzy et al from 1991.

← Previous revision		Revision as of 08:31, 9 January 2023
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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.		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.		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. [http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=119727&isnumber=3425]		WTMM was originally developed by Mallat and Hwang in 1992 and used for image processing [http://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.		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.

Qwerfjkl (bot): Capitalising short description "method for detecting a signal's fractal dimension" per WP:SDFORMAT (via Bandersnatch)

2022-02-08T21:10:42Z

Capitalising short description "method for detecting a signal's fractal dimension" per WP:SDFORMAT (via Bandersnatch)

← Previous revision		Revision as of 21:10, 8 February 2022
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	{{Short description\|~~method~~ for detecting a signal's fractal dimension}}		{{Short description\|Method for detecting a signal's fractal dimension}}
	The '''wavelet transform modulus maxima (WTMM)''' is a method for detecting the [[fractal dimension]] of a signal.		The '''wavelet transform modulus maxima (WTMM)''' is a method for detecting the [[fractal dimension]] of a signal.

Cherno More at 16:25, 30 January 2022

2022-01-30T16:25:36Z

← Previous revision		Revision as of 16:25, 30 January 2022
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	: <math>f(t) = a_0 + a_1 (t - t_i) + a_2(t - t_i)^2 + \cdots + a_h(t - t_i)^{h_i} \, </math>		: <math>f(t) = a_0 + a_1 (t - t_i) + a_2(t - t_i)^2 + \cdots + a_h(t - t_i)^{h_i} \, </math>

	where <math> t </math> is close to <math> t_i </math> and <math> h_i </math> is a non-integer quantifying the local singularity. (Compare this to a [[Taylor series]], where in practice only a limited number of low-order terms are used to approximate a continuous function.)		where <math> t </math> is close to <math> t_i </math> and <math> h_i </math> is a non-integer quantifying the local singularity. (Compare this to a [[Taylor series]], where in practice only a limited number of low-order terms are used to approximate a continuous function.)

	Generally, a [[continuous wavelet transform]] decomposes a signal as a function of time, rather than assuming the signal is stationary (For example, the Fourier transform). Any continuous wavelet can be used, though the first derivative of the [[Gaussian distribution]] and the [[Mexican hat wavelet]] (2nd derivative of Gaussian) are common. Choice of wavelet may depend on characteristics of the signal being investigated.		Generally, a [[continuous wavelet transform]] decomposes a signal as a function of time, rather than assuming the signal is stationary (For example, the Fourier transform). Any continuous wavelet can be used, though the first derivative of the [[Gaussian distribution]] and the [[Mexican hat wavelet]] (2nd derivative of Gaussian) are common. Choice of wavelet may depend on characteristics of the signal being investigated.

	Below we see one possible wavelet basis given by the first derivative of the Gaussian:		Below we see one possible wavelet basis given by the first derivative of the Gaussian:
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	: <math>G' (t,a,b) = \frac{a}{(2\pi)^{-1/2}}(t - b) e^{\left(\frac{-(t-b)^2}{2a^2}\right)} \,</math>		: <math>G' (t,a,b) = \frac{a}{(2\pi)^{-1/2}}(t - b) e^{\left(\frac{-(t-b)^2}{2a^2}\right)} \,</math>

	Once a "mother wavelet" is chosen, the continuous wavelet transform is carried out as a continuous, [[square-integrable function]] that can be scaled and translated. Let <math>a > 0</math> be the scaling constant and <math>b\in\mathbb{R}</math> be the translation of the wavelet along the signal:		Once a "mother wavelet" is chosen, the continuous wavelet transform is carried out as a continuous, [[square-integrable function]] that can be scaled and translated. Let <math>a > 0</math> be the scaling constant and <math>b\in\mathbb{R}</math> be the translation of the wavelet along the signal:

	: <math>X_w(a,b)=\frac{1}{\sqrt{a}} \int_{-\infty}^\infty x(t)\psi^\ast \left(\frac{t-b}{a}\right)\, dt</math>		: <math>X_w(a,b)=\frac{1}{\sqrt{a}} \int_{-\infty}^\infty x(t)\psi^\ast \left(\frac{t-b}{a}\right)\, dt</math>

	where <math>\psi(t)</math> is a continuous function in both the time domain and the frequency domain called the mother wavelet and <math>^{\ast}</math> represents the operation of [[complex conjugate]].		where <math>\psi(t)</math> is a continuous function in both the time domain and the frequency domain called the mother wavelet and <math>^{\ast}</math> represents the operation of [[complex conjugate]].

	By calculating <math>X_w(a,b) </math> for subsequent wavelets that are derivatives of the mother wavelet, singularities can be identified. Successive derivative wavelets remove the contribution of lower order terms in the signal, allowing the maximum <math>h_i</math> to be detected. (Recall that when taking derivatives, lower order terms become 0.) This is the "modulus maxima".		By calculating <math>X_w(a,b) </math> for subsequent wavelets that are derivatives of the mother wavelet, singularities can be identified. Successive derivative wavelets remove the contribution of lower order terms in the signal, allowing the maximum <math>h_i</math> to be detected. (Recall that when taking derivatives, lower order terms become 0.) This is the "modulus maxima".

	Thus, this method identifies the singularity spectrum by convolving the signal with a wavelet at different scales and time offsets.		Thus, this method identifies the singularity spectrum by convolving the signal with a wavelet at different scales and time offsets.

Shd33: removed repetition

2021-03-17T14:08:43Z

removed repetition

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	More than this, the WTMM is capable of partitioning the time and scale domain of a signal into fractal dimension regions, and the method is sometimes referred to as a "mathematical microscope" due to its ability to inspect the multi-scale dimensional characteristics of a signal and possibly inform about the sources of these characteristics.		More than this, the WTMM is capable of partitioning the time and scale domain of a signal into fractal dimension regions, and the method is sometimes referred to as a "mathematical microscope" due to its ability to inspect the multi-scale dimensional characteristics of a signal and possibly inform about the sources of these characteristics.

	The WTMM method uses [[continuous wavelet transform]] rather than [[Fourier transform]]s to detect ~~singularities~~ [[Mathematical singularity\|~~singularity~~]] – that is discontinuities, areas in the signal that are not continuous at a particular derivative.		The WTMM method uses [[continuous wavelet transform]] rather than [[Fourier transform]]s to detect [[Mathematical singularity\|singularities]] – that is discontinuities, areas in the signal that are not continuous at a particular derivative.

	In particular, this method is useful when analyzing [[multifractal]] signals, that is, signals having multiple fractal dimensions.		In particular, this method is useful when analyzing [[multifractal]] signals, that is, signals having multiple fractal dimensions.

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← Previous revision		Revision as of 20:25, 2 March 2020
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	* 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}} [http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=119727&isnumber=3425]		* 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}} [http://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]		* Arneodo on Wavelets [http://www.iscpif.fr/tiki-index.php?page=CSSS'08+Arneodo&highlight=towards]
	* {{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 \| pages=3515–3518}}		* {{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 \| pages=875–884}}		* {{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 }}

	[[Category:Wavelets]]		[[Category:Wavelets]]

Forbes72: /* References */ improve existing refs

2020-01-15T03:58:52Z

References: improve existing refs

← Previous revision		Revision as of 03:58, 15 January 2020
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	* 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}} [http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=119727&isnumber=3425]		* 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}} [http://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]		* Arneodo on Wavelets [http://www.iscpif.fr/tiki-index.php?page=CSSS'08+Arneodo&highlight=towards]
	* "Wavelets and multifractal formalism for singular signals : ~~application~~ to turbulence data", ~~J.F.~~ ~~Muzy,~~ E. ~~Bacry~~ ~~and~~ ~~A. Arneodo,~~ ''Physical ~~Review~~ ~~Letters''~~ 67, ~~3515~~ (1991). ~~[http:~~/~~/prl~~.~~aps~~.~~org/abstract/PRL/v67/i25/p3515_1]~~		* {{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 \| pages=3515–3518}}
	* "Multifractal formalism for fractal signals: ~~the~~ structure function approach versus the wavelet transform modulus maxima method", ~~J.F.~~ ~~Muzy,~~ E. ~~Bacry~~ ~~and~~ A. ~~Arneodo,~~ ~~''Phys.~~ ~~Rev. E''~~ 47, ~~875~~ ~~[http:~~/~~/pre~~.~~aps~~.~~org/abstract/PRE/v47/i2/p875_1]~~		* {{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 \| pages=875–884}}

	[[Category:Wavelets]]		[[Category:Wavelets]]

Daviddwd: short description

2019-10-19T03:02:49Z

short description

← Previous revision		Revision as of 03:02, 19 October 2019
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			{{Short description\|method for detecting a signal's fractal dimension}}
	The '''wavelet transform modulus maxima (WTMM)''' is a method for detecting the [[fractal dimension]] of a signal.		The '''wavelet transform modulus maxima (WTMM)''' is a method for detecting the [[fractal dimension]] of a signal.

Chongkian: /* References */

2019-09-14T14:03:22Z

References

← Previous revision		Revision as of 14:03, 14 September 2019
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	* "Multifractal formalism for fractal signals: the structure function approach versus the wavelet transform modulus maxima method", J.F. Muzy, E. Bacry and A. Arneodo, ''Phys. Rev. E'' 47, 875 [http://pre.aps.org/abstract/PRE/v47/i2/p875_1]		* "Multifractal formalism for fractal signals: the structure function approach versus the wavelet transform modulus maxima method", J.F. Muzy, E. Bacry and A. Arneodo, ''Phys. Rev. E'' 47, 875 [http://pre.aps.org/abstract/PRE/v47/i2/p875_1]

	[[Category:Wavelets\| ]]		[[Category:Wavelets]]