Wavelet

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 For practical applications one prefers for efficiency reasons continuously differentiable functions with compact support as mother (prototype) wavelet (functions). However, to satisfy analytical requirements (in the continuous WT) and in general for theoretical reasons one chooses the wavelet functions from a subspace of the Space $L^1(\R)\cap L^2(\R)$ . This is the space of Measurable Functions that are both absolutely and square Integrable :
	:<math>\int {-\infty}^{\infty} \psi (t)^2\, Dt	1</math> is the condition for square norm one

For the continuous WT, the pair ''(a,b)'' varies over the full half-plane

\R_+	imes\R

; for the discrete WT this pair varies over a discrete subset of it, which is also called ''affine group''.

These functions are often incorrectly referred to as the basis functions of the (continuous) transform. In fact, as in the continuous Fourier transform, there is no basis in the continuous wavelet transform. 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 discrete 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.

Daubechies and Symlet wavelets can be defined by the scaling filter.

Scaling function

Wavelets are 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 {Link without Title} 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''.

Meyer wavelets can be defined by scaling functions

Wavelet function

The wavelet only has a time domain representation as the wavelet function

\psi (t)

.

Mexican Hat Wavelet s can be defined by a wavelet function.

APPLICATIONS

Generally, the DWT is used for Source 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 , Nick Kingsbury

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: