Wavelet transform modulus maxima method

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

In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer.

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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 transform rather than Fourier transforms to detect singularities singularity – that is discontinuities, areas in the signal that are not continuous at a particular derivative.

In mathematics, the continuous wavelet transform (CWT) is a formal tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of the wavelets vary continuously.

The Fourier transform (FT) decomposes a function of time into its constituent frequencies. This is similar to the way a musical chord can be expressed in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform of a function of time is itself a complex-valued function of frequency, whose magnitude (modulus) represents the amount of that frequency present in the original function, and whose argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain. There is also an inverse Fourier transform that mathematically synthesizes the original function from its frequency domain representation.

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

Description

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

${\displaystyle f(t)=a_{0}+a_{1}(t-t_{i})+a_{2}(t-t_{i})^{2}+\cdots +a_{h}(t-t_{i})^{h_{i}}\,}$

where ${\displaystyle t}$ is close to ${\displaystyle t_{i}}$ and ${\displaystyle h_{i}}$ 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.)

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

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.

In mathematics and numerical analysis, the Ricker wavelet

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

${\displaystyle G'(t,a,b)={\frac {a}{(2\pi )^{-1/2}}}(t-b)e^{\left({\frac {-(t-b)^{2}}{2a^{2}}}\right)}\,}$

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 ${\displaystyle a>0}$ be the scaling constant and ${\displaystyle b\in \mathbb {R} }$ be the translation of the wavelet along the signal:

In mathematics, a square-integrable function, also called a quadratically integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line is defined as follows.

${\displaystyle X_{w}(a,b)={\frac {1}{\sqrt {a}}}\int _{-\infty }^{\infty }x(t)\psi ^{\ast }\left({\frac {t-b}{a}}\right)\,dt}$

where ${\displaystyle \psi (t)}$ is a continuous function in both the time domain and the frequency domain called the mother wavelet and ${\displaystyle ^{\ast }}$ represents the operation of complex conjugate.

By calculating ${\displaystyle X_{w}(a,b)}$ 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 ${\displaystyle h_{i}}$ 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.

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

History

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.