Wavelet

Wavelets, wavelet analysis, and the wavelet transform refers to the representation of a signal in terms of a finite length or fast decaying oscillating waveform (known as the mother wavelet). This waveform is scaled and translated to match the input signal.

Overview

The word wavelet is due to Morlet and Grossman in the early 1980s. They used the French word ondelette - meaning "small wave". A little later it was transformed into English by translating "onde" into "wave" - giving wavelet. Wavelet transforms are broadly classified into the discrete wavelet transform (DWT) and the continuous wavelet transform (CWT). The principal difference between the two is the continuous transform operates over every possible scale and translation whereas the discrete uses a specific subset of all scale and translation values.

Using wavelet theory

Wavelet theory is applicable to several other subjects. All wavelet transforms may be considered to be forms of time-frequency representation and are, therefore, related to the subject of harmonic analysis. Almost all practically useful discrete wavelet transforms make use of filterbanks containing finite impulse response filters. The wavelets forming a CWT are subject to Heisenberg's uncertainty principle and, equivalently, discrete wavelet bases may be considered in the context of other forms of the uncertainty principle.

Mother wavelet

Although one prefers continuously differentiable functions with compact support as mother (prototype) wavelet (functions), the most general conditions are that it is a function in the space $L^{1}(\mathbb {R} )\cap L^{2}(\mathbb {R} )$ with zero mean and square norm one (plus additional conditions depending on the type of the transform). Those conditions translate into the existence of the following integrals:

$\int _{-\infty }^{\infty }|\psi (t)|\ ^{2}\,dt=1$, i.e. $\psi \in L^{2}(\mathbb {R} )$ and normalized

$\int _{-\infty }^{\infty }|\psi \ (t)|\,dt<\infty$, i.e. $\psi \in L^{1}(\mathbb {R} )$ and

$\int _{-\infty }^{\infty }\psi \ (t)\,dt=0$, i.e. zero mean.

In most situations it is useful to demand that $\psi$ be continuous and has a higher number M of vanishing moments, i.e. for all integer m<M

$\int _{-\infty }^{\infty }t^{m}\,\psi \ (t)\,dt=0$

For the continuous WT, the mother wavelet must satisfy an admissibility criterion (loosely speaking, a kind of half-differentiability) in order to get a stably invertible transform. For the discrete WT, one needs the existence of a father wavelet and corresponding multiresolution analysis, from which additional algebraic conditions ensue.

Some example mother wavelets are:

Meyer

Morlet

Mexican Hat

The mother wavelet is scaled (or dilated) by a factor of $a$ and translated (or shifted) by a factor of $b$ to give (under Morlet's original formulation):

$\psi _{a,b}(t)={1 \over {\sqrt {a}}}\psi \left({{t-b} \over a}\right)$

These functions are often incorrectly referred to as the basis functions of the transform. In fact, there is no basis. Time-frequency interpretation uses a subtly different formulation (after Delprat).

Comparisons with Fourier

The wavelet transform is often compared with the Fourier transform, in which signals are represented as a sum of sinusoids. The main difference is that wavelets are localized in both time and frequency whereas the standard Fourier transform is only localized in frequency. The Short-time Fourier transform (STFT) is also time and frequency localized but there are issues with the frequency time resolution and wavelets often give a better signal representation using Multiresolution analysis.

The wavelet transform is also less computationally complex, taking O(N) time as compared to O(N log N) for the fast Fourier transform (N is the data size).

Definition of a wavelet

There are a number of ways of defining a wavelet (or a wavelet family).

Scaling filter

The wavelet is entirely defined by the scaling filter g - a low-pass finite impulse response (FIR) filter of length 2N and sum 1. In biorthogonal wavelets, separate decomposition and reconstruction filters are defined.

For analysis the high pass filter is calculated as the QMF of the low pass, and reconstruction filters the time reverse of the decomposition.

e.g. Daubechies and Symlet wavelets

Scaling function

Wavelet defined by the wavelet function $\psi (t)$ (i.e. the mother wavelet) and scaling function $\phi (t)$ (also called father wavelet) in the time domain.

The wavelet function is in effect a band-pass filter and scaling it for each level halves its bandwidth. This creates the problem that in order to cover the entire spectrum an infinite number of levels would be required. The scaling function filters the lowest level of the transform and ensures all the spectrum is covered. See [1] for a detailed explanation.

For a wavelet with compact support, $\phi (t)$ can be considered finite in length and is equivalent to the scaling filter g.

e.g. Meyer wavelet

Wavelet function

The wavelet only has a time domain representation as the wavelet function $\psi (t)$.

e.g. Mexican hat wavelet

Applications

Generally, the DWT is used for signal coding whereas the CWT is used for signal analysis. Consequently, the DWT is commonly used in engineering and computer science and the CWT is most often used in scientific research. Wavelet transforms are now being adopted for a vast number of different applications, often replacing the conventional Fourier transform. Many areas of physics have seen this paradigm shift, including molecular dynamics, ab initio calculations, astrophysics, density-matrix localisation, seismic geophysics, optics, turbulence and quantum mechanics. Other areas seeing this change have been image processing, blood-pressure, heart-rate and ECG analyses, DNA analysis, protein analysis, climatology, general signal processing, speech recognition, computer graphics and multifractal analysis.

One use of wavelets is in data compression. Like several other transforms, the wavelet transform can be used to transform raw data (like images), then encode the transformed data, resulting in effective compression. JPEG 2000 is an image standard that uses wavelets. For details see wavelet compression.

History

The development of wavelets can be linked to several separate trains of thought, starting with Haar's work in the early 20th century. Notable contributions to wavelet theory can be attributed to Goupillaud, Grossman and Morlet's formulation of what is now known as the CWT (1982), Strömberg's early work on discrete wavelets (1983), Daubechies' orthogonal wavelets with compact support (1988), Mallat's multiresolution framework (1989), Delprat's time-frequency interpretation of the CWT (1991), Newland's Harmonic wavelet transform and many others since.

Time line

• First wavelet (Haar wavelet) by Alfred Haar (1909)
• Since the 1950s: Jean Morlet and Alex Grossman
• Since the 1980s: Yves Meyer, Stéphane Mallat, Ingrid Daubechies, Ronald Coifman, Victor Wickerhauser

Wavelet transforms

There are a large number of wavelet transforms each suitable for different applications. For a full list see list of wavelet-related transforms but the common ones are listed below:

• Continuous wavelet transform (CWT)
• Discrete wavelet transform (DWT)
• Fast wavelet transform (FWT)
• Wavelet packet decomposition (WPD)

List of wavelets

Discrete wavelets

• Beylkin (18)
• Coiflet (6, 12, 18, 24, 30)
• Daubechies wavelet (2, 4, 6, 8, 10, 12, 14, 16, 18, 20)
• Cohen-Daubechies-Feauveau wavelet (Sometimes referred to as Daubechies biorthogonal, bior44=CDF9/7)
• Haar wavelet
• Vaidyanathan filter (24)
• Symmlet
• Complex wavelet transform

Continuous wavelets

• Mexican hat wavelet
• Hermitian wavelet
• Hermitian hat wavelet
• Complex mexican hat wavelet
• Morlet wavelet
• Modified Morlet wavelet
• Addison wavelet
• Hilbert-Hermitian wavelet

See also

• Filter banks
• Scale space
• Ultra wideband radio- transmits wavelets.
• short-time Fourier transform
• chirplet transform
• fractional Fourier transform

References

• Paul S. Addison, The Illustrated Wavelet Transform Handbook, Institute of Physics, 2002, ISBN 0750306920
• Ingrid Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, 1992, ISBN 0898712742
• Mladen Victor Wickerhauser, Adapted Wavelet Analysis From Theory to Software, A K Peters Ltd, 1994, ISBN 1568810415
• P. P. Viadyanathan, Multirate Systems and Filter Banks, Prentice Hall, 1993, ISBN 0136057187

External links

• Wavelet Digest
• Amaras Wavelet Page
• Wavelet Posting Board
• The Wavelet Tutorial by Polikar
• OpenSource Wavelet C Code
• An Introduction to Wavelets
• Filter Coefficients of Popular Wavelets
• Wavelet-based time-frequency analysis in Mathematica and example analyses from physics, biology, engineering, bioinformatics and finance.
• Wavelets for Kids (PDF file) (introductory)
• Link collection about wavelets
• List of Wavelet resources, libraries and source codes
• A really friendly guide to wavelets
• Wavelet forums (French) Wavelet forum (English)