Wavelet transform modulus maxima method: Difference between revisions

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[[Category:Wavelets| ]]

The '''Waveletwavelet Transformtransform Modulusmodulus Maximamaxima (WTMM)''' is a method for detecting the [[fractal dimension]] of a signal.

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.

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

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Consider a signal that can be represented by the following equation:

: <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.)

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Below we see one possible wavelet basis given by the first derivative of the Gaussian:

: <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:

: <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]].

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

The WTMM is then capable of producing a "skeleton" that partitions the scale and time space by fractal dimension.

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== References ==

* 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 : 0-12-466606-X; Academic Press, 1999[http://www.ceremade.dauphine.fr/~peyre/wavelet-tour/]

* Mallat, S.; Hwang, W.L.; , "Singularity detection and processing with wavelets," Information Theory, IEEE Transactions on , vol.38, no.2, pp.617-643617–643, 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]

* Wavelets and multifractal formalism for singular signals : application to turbulence data

J.F. Muzy, E. Bacry and A. Arneodo, Phys. Rev. Lett. 67, 3515 (1991). [http://prl.aps.org/abstract/PRL/v67/i25/p3515_1]

* Multifractal formalism for fractal signals: the structure fonction 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]