Wavelet transform modulus maxima method: Difference between revisions

Revision as of 15:01, 12 April 2011

Revision as of 15:03, 12 April 2011

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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.

The WTMM method uses [[continuous_wavelet_transformcontinuous wavelet transform]] rather than [[Fourier_transformFourier transform]]s to detect singularities[[[Mathematical_singularityMathematical 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 [[Multifractalmultifractal]] signals, that is, signals having multiple fractal dimensions.

== Description ==

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<math>f(t) = a_0 + a_1 (t - t_i) + a_2(t - t_i)^2 + ... + 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 [http://en.wikipedia.org/wiki/[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 [http://en.wikipedia.org/wiki/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]] and the mexican hat wavelet[[Mexican_hat_wavelet]] (2nd derivative of Gaussian)[http://en.wikipedia.org/wiki/Mexican_hat_wavelet] 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: