NationMaster - Encyclopedia: Wavelet

A wavelet is a kind of mathematical function used to divide a given function into different frequency components and study each component with a resolution that matches its scale. A wavelet transform is the representation of a function by wavelets. The wavelets are scaled and translated copies (known as "daughter wavelets") of a finite-length or fast-decaying oscillating waveform (known as the "mother wavelet"). Wavelet transforms have advantages over traditional Fourier transforms for representing functions that have discontinuities and sharp peaks. In Euclidean geometry, uniform scaling is a linear transformation that enlarges or diminishes objects; the scale factor is the same in all directions; it is also called a homothety. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...

In formal terms, this representation is a wavelet series representation of a square integrable function with respect to either a complete, orthonormal set of basis functions, or an overcomplete set of frame functions (also known as a Riesz basis), for the Hilbert space of square integrable functions. In mathematics, a wavelet series is a representation of a square-integrable (real or complex valued) function by a certain orthonormal series generated by a wavelet. ... In mathematics, the term integrable function refers to a function whose integral may be calculated. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In linear algebra, two vectors v and w are said to be orthonormal if they are both orthogonal (according to a given inner product) and normalized. ... In functional analysis and its applications, a function space can be viewed as a vector space of infinite dimension whose basis vectors are functions not vectors. ... In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. ... A sequence of vectors in a Hilbert space is called a Riesz sequence if there exist constants such that for all sequences of scalars . ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...

Wavelet transforms are classified into discrete wavelet transforms (DWTs) and continuous wavelet transforms (CWTs). CWTs operate over every possible scale and translation whereas DWTs use a specific subset of scale and translation values. In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. ... // Formulation In mathematics and signal processing, the continuous wavelet transform (CWT) of a function is a wavelet transform defined by where represents translation, represents scale and is the mother wavelet. ...

Look up wavelet in Wiktionary, the free dictionary.

The word wavelet is due to Morlet and Grossmann in the early 1980s. They used the French word ondelette, meaning "small wave". Soon it was transferred to English by translating "onde" into "wave", giving "wavelet". Wikipedia does not have an article with this exact name. ... Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 150 languages. ... Jean Morlet is a French geophysicist who did pioneering work in the field of wavelet analysis in collaboration with Alex Grossman. ... Alex Grossman is a Croatian physicist at the University of Marseilles who did pioneering work on wavelet analysis with Jean Morlet. ... This article cites very few or no references or sources. ...

Wavelet theory

Wavelet theory is very applicable to several other subjects. All wavelet transforms may be considered forms of time-frequency representation and so are related to harmonic analysis. Almost all practically useful discrete wavelet transforms use 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. A time-frequency representation (TFR) is a view of a signal (taken to be a function of time) represented over both time and frequency. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... A finite impulse response (FIR) filter is a type of a digital filter. ... Werner Karl Heisenberg (December 5, 1901 â€“ February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics, and acknowledged to be one of the most important physicists of the twentieth century. ... In quantum physics, the Heisenberg uncertainty principle is a mathematical property of a pair of canonical conjugate quantities - usually stated in a form of reciprocity of spans of their spectra. ...

Wavelet transforms are broadly divided into three classes: continuous, discretised and multiresolution-based.

Continuous wavelet transforms

In continuous wavelet transforms, a given signal of finite energy is projected on a continuous family of frequency bands (or similar subspaces of the function space $L^2(R)$ ), for instance on every frequency band of the form $[f,2 f]$ for all positive frequencies f>0. Then, the original signal can be reconstructed by a suitable integration over all the resulting frequency components. // Formulation In mathematics and signal processing, the continuous wavelet transform (CWT) of a function is a wavelet transform defined by where represents translation, represents scale and is the mother wavelet. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ...

The frequency bands or subspaces are scaled versions of a subspace at scale 1. This subspace in turn is in most situations generated by the shifts of one generating function $psi in L^2(R)$ , the mother wavelet. For the example of the scale one frequency band $[1,2]$ this function is

$psi(t)=2,operatorname{sinc}(2t)-,operatorname{sinc}(t)=frac{sin(2pi t)-sin(pi t)}{pi t}$

with the (normalized) sinc function. Other example mother wavelets are: The sinc function sinc(x) from x = âˆ’8Ï€ to 8Ï€. In mathematics, the sinc function (for sinus cardinalis), also known as the interpolation function, filtering function or the first spherical Bessel function , is the product of a sine function and a monotonically decreasing function. ...

Meyer

Morlet

Mexican Hat

The subspace of scale a or frequency band $[1/a,,2/a]$ is generated by the functions (sometimes called child wavelets) Image File history File links The meyer wavelet. ... Image File history File links The meyer wavelet. ... Image File history File links The morlet wavelet. ... Image File history File links The morlet wavelet. ... Image File history File links The mexican hat wavelet. ... Image File history File links The mexican hat wavelet. ...

$psi_{a,b} (t) = frac1{sqrt a }psi left( frac{t - b}{a} right)$ ,

where a is positive and defines the scale and b is any real number and defines the shift. The pair (a,b) defines a point in the upper halfplane $R_+times R$ .

The projection of a function x onto the subspace of scale a then has the form

$x_a(t)=int_R WT_psi{x}(a,b)cdotpsi_{a,b}(t),db$

with wavelet coefficients

$WT_psi{x}(a,b)=langle x,psi_{a,b}rangle=int_R x(t)overline{psi_{a,b}(t)},dt$ .

For the analysis of the signal x, one can assemble the wavelet coefficients into a scaleogram of the signal. Scaleograms from the DWT and CWT for an audio sample In signal processing, a scaleogram is a visual method of displaying a wavelet transform. ...

Discrete wavelet transforms

It is computationally impossible to analyze a signal using all wavelet coefficients. So one may wonder if it is sufficient to pick a discrete subset of the upper halfplane to be able to reconstruct a signal from the corresponding wavelet coefficients. One such system is the affine system for some real parameters a>1, b>0. The corresponding discrete subset of the halfplane consists of all the points $(a^m, n,a^m b)$ with integers $m,ninZ$ . The corresponding baby wavelets are now given as

ψ m, n (t) = a - m / 2 ψ(a - m t - n b)

A sufficient condition for the reconstruction of any signal x of finite energy by the formula

$x(t)=sum_{minZ}sum_{ninZ}langle x,,psi_{m,n}ranglecdotpsi_{m,n}(t)$

is that the functions ${psi_{m,n}:m,ninZ}$ form a tight frame of $L^2(R)$ . In mathematics, a frame of a vector space V with a scalar product can be seen as a generalization of the idea of a basis to sets which are linearly dependent. ...

MRA-based discrete wavelet transforms

D4 wavelet

In any discretised wavelet transform, there are only a finite number of wavelet coefficients for each bounded rectangular region in the upper halfplane. Still, each coefficient requires the evaluation of an integral. To avoid this numerical complexity, one needs one auxiliary function, the father wavelet $phiin L^2(R)$ . Further, one has to restrict a to be an integer. A typical choice is a=2 and b=1. The most famous pair of father and mother wavelets is the Daubechies 4 tap wavelet. Image File history File links Please see the file description page for further information. ... Image File history File links Please see the file description page for further information. ... Daubechies 20 2-d wavelet (Wavelet Fn X Scaling Fn) Named after Ingrid Daubechies, the Daubechies wavelets are a family of orthogonal wavelets defining a discrete wavelet transform and characterized by a maximal number of vanishing moments for some given support. ...

From the mother and father wavelets one constructs the subspaces

$V_m=operatorname{span}(phi_{m,n}:ninZ)$ , where

φ m, n (t) = 2 - m / 2 φ(2 - m t - n)

and

$W_m=operatorname{span}(psi_{m,n}:ninZ)$ , where

ψ m, n (t) = 2 - m / 2 ψ(2 - m t - n)

From these one requires that the sequence

${0}subsetdotssubset V_1subset V_0subset V_{-1}subsetdotssubset L^2(R)$

forms a multiresolution analysis of $L^2(R)$ and that the subspaces $dots,W_1,W_0,W_{-1},dotsdots$ are the orthogonal "differences" of the above sequence, that is, $W m$ is the orthogonal complement of $V m$ inside the subspace $V m - 1$ . In analogy to the sampling theorem one may conclude that the space $V m$ with sampling distance $2 m$ more or less covers the frequency baseband from 0 to $2 - m - 1$ . As orthogonal complement, $W m$ roughly covers the band $[2 - m - 1,2 - m]$ . A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant discrete wavelet transforms (DWT) and the justification for the algorithm of the fast wavelet transform (FWT). ... The Nyquist-Shannon sampling theorem is the fundamental theorem in the field of information theory, in particular telecommunications. ...

From those inclusions and orthogonality relations follows the existence of sequences $h={h_n}_{ninZ}$ and $g={g_n}_{ninZ}$ that satisfy the identities

$h_n=langlephi_{0,0},,phi_{1,n}rangle$ and $phi(t)=sqrt2 sum_{ninZ} h_nphi(2t-n)$

and

$g_n=langlepsi_{0,0},,phi_{1,n}rangle$ and $psi(t)=sqrt2 sum_{ninZ} g_nphi(2t-n)$ .

The second identity of the first pair is a refinement equation for the father wavelet $φ$ . Both pairs of identities form the basis for the algorithm of the fast wavelet transform. In mathematics, in the area of wavelet analysis, a refinable function is a function which fulfills some kind of self-similarity. ... The Fast (Lifting) Wavelet Transform is a mathematical algorithm designed to turn a waveform or signal in the time domain into a sequence of coefficients based on an orthogonal basis of small finite waves, or wavelets. ...

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 absolutely and square integrable: In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... The integral can be interpreted as the area under a curve. ... In mathematics, the term integrable function refers to a function whose integral may be calculated. ...

$int_{-infty}^{infty} |psi (t)|, dt <infty$ and $int_{-infty}^{infty} |psi (t)|^2 , dt <infty$ .

Being in this space ensures that one can formulate the conditions of zero mean and square norm one:

$int_{-infty}^{infty} psi (t), dt = 0$ is the condition for zero mean, and

$int_{-infty}^{infty} |psi (t)|^2, dt = 1$ is the condition for square norm one.

For $ψ$ to be a wavelet for the continuous wavelet transform (see there for exact statement), the mother wavelet must satisfy an admissibility criterion (loosely speaking, a kind of half-differentiability) in order to get a stably invertible transform. // Formulation In mathematics and signal processing, the continuous wavelet transform (CWT) of a function is a wavelet transform defined by where represents translation, represents scale and is the mother wavelet. ...

For the discrete wavelet transform, one needs at least the condition that the wavelet series is a representation of the identity in the space $L^2(R)$ . Most constructions of discrete WT make use of the multiresolution analysis, which defines the wavelet by a scaling function. This scaling function itself is solution to a functional equation. In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. ... In mathematics, a wavelet series is a representation of a square-integrable (real or complex valued) function by a certain orthonormal series generated by a wavelet. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant discrete wavelet transforms (DWT) and the justification for the algorithm of the fast wavelet transform (FWT). ...

In most situations it is useful to restrict $ψ$ to be a continuous function with a higher number M of vanishing moments, i.e. for all integer m<M

$int_{-infty}^{infty} t^m,psi (t), dt = 0$

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): Image File history File links The meyer wavelet. ... Image File history File links The meyer wavelet. ... Image File history File links The morlet wavelet. ... Image File history File links The morlet wavelet. ... Image File history File links The mexican hat wavelet. ... Image File history File links The mexican hat wavelet. ...

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

For the continuous WT, the pair (a,b) varies over the full half-plane $R_+timesR$ ; 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. In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ... The short-time Fourier transform (STFT), or short-term Fourier transform, is a Fourier-related transform used to determine the sinusoidal frequency and phase content of a signal as it changes over time. ... A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant discrete wavelet transforms (DWT) and the justification for the algorithm of the fast wavelet transform (FWT). ...

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). Complexity in general usage is the opposite of simplicity. ... The Fast Fourier Transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. ...

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. A finite impulse response (FIR) filter is a type of a digital filter. ...

For analysis the high pass filter is calculated as the quadrature mirror filter of the low pass, and reconstruction filters the time reverse of the decomposition. In digital signal processing, a quadrature mirror filter is a filter bank which splits an input signal into two bands which are usually then subsampled by a factor of 2. ...

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

Scaling function

Wavelets are defined by the wavelet function $ψ(t)$ (i.e. the mother wavelet) and scaling function $φ(t)$ (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, $φ(t)$ 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 $ψ(t)$ .

Mexican hat wavelets can be defined by a wavelet function. In mathematics and numerical analysis, the Mexican hat wavelet is the normalized, second derivative of a Gaussian function. ...

Applications

Generally, the DWT is used for data compression, and the CWT for signal analysis. Thus, the DWT is commonly used in engineering and computer science, and the CWT in scientific research. In computer science and information theory, data compression or source coding is the process of encoding information using fewer bits (or other information-bearing units) than an unencoded representation would use through use of specific encoding schemes. ... Signal analysis is the extraction of information from a signal. ...

Wavelet transforms are now being adopted for a vast number of 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. This change has also occurred in image processing, blood-pressure, heart-rate and ECG analyses, DNA analysis, protein analysis, climatology, general signal processing, speech recognition, computer graphics and multifractal analysis. In computer vision and image processing, the notion of scale-space representation and Gaussian derivative operators is regarded as a canonical multi-scale representation. In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... Molecular dynamics (MD) is a form of computer simulation where atoms and molecules are allowed to interact for a period of time under known laws of physics. ... In the physical sciences, an ab initio calculation is one performed from first principles. ... Spiral Galaxy ESO 269-57 Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties (luminosity, density, temperature, and chemical composition) of celestial objects such as stars, galaxies, and the interstellar medium, as well as their interactions. ... A density matrix is a self-adjoint (or Hermitian) positive-semidefinite matrix, (possibly infinite dimensional), of trace one, that describes the statistical state of a quantum system. ... For the book by Sir Isaac Newton, see Opticks. ... In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic, stochastic property changes. ... Fig. ... UPIICSA IPN - Binary image Image processing is any form of information processing for which the input is an image, such as photographs or frames of video; the output is not necessarily an image, but can be for instance a set of features of the image. ... ECG may also refer to the East Coast Greenway Lead II An Electrocardiogram (ECG or EKG, abbreviated from the German Elektrokardiogramm) is a graphic produced by an electrocardiograph, which records the electrical voltage in the heart in the form of a continuous strip graph. ... The structure of part of a DNA double helix Deoxyribonucleic acid (DNA) is a nucleic acid that contains the genetic instructions for the development and function of living organisms. ... A representation of the 3D structure of myoglobin, showing coloured alpha helices. ... Climatology is the study of climate, scientifically defined as weather conditions averaged over a period of time,[1] and is a branch of the atmospheric sciences. ... Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... Speech recognition (in many contexts also known as automatic speech recognition, computer speech recognition or erroneously as Voice Recognition) is the process of converting a speech signal to a sequence of words, by means of an algorithm implemented as a computer program. ... Computer graphics is a sub-field of computer science and is concerned with digitally synthesizing and manipulating visual content. ... In mathematical analysis, multifractal analysis is the process of determining the fractal dimension of a multifractal system. ... Computer vision is the science and technology of machines that see. ... UPIICSA IPN - Binary image Image processing is any form of information processing for which the input is an image, such as photographs or frames of video; the output is not necessarily an image, but can be for instance a set of features of the image. ... Scale space theory is a framework for multi-scale signal representation developed by the computer vision and image processing communities. ...

One use of wavelets is in data compression. Like some other transforms, wavelet transforms can be used to transform data, then encode the transformed data, resulting in effective compression. For example, JPEG 2000 is an image compression standard that uses biorthogonal wavelets. This means that although the frame is overcomplete, it is a tight frame, and the same frame functions (except for conjugation in the case of complex wavelets) are used for both analysis and synthesis, i.e., in both the forward and inverse transform. For details see wavelet compression. JPEG 2000 is a wavelet-based image compression standard. ... Wavelet compression is a form of data compression well suited for image compression (sometimes also video compression and audio 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 Zweig’s discovery of the continuous wavelet transform in 1975 (originally called the cochlear transform and discovered while studying the reaction of the ear to sound)[2], Goupillaud, Grossmann 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. Alfr d Haar (October 11, 1885 - March 16, 1933) was a Hungarian mathematician. ... George Zweig was originally trained as a particle physicist under Richard Feynman and later turned his attention to neurobiology. ... Alex Grossman is a Croatian physicist at the University of Marseilles who did pioneering work on wavelet analysis with Jean Morlet. ... Jean Morlet is a French geophysicist who did pioneering work in the field of wavelet analysis in collaboration with Alex Grossman. ... Ingrid Daubechies (born August 17, 1954) is a Belgian physicist and mathematician. ... Stéphane Mallat Stéphane G. Mallat made some fundamental contributions to the development of wavelet theory in the late 1980s and early 1990s. ...

Time line

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

The Haar wavelet The Haar wavelet is the first known wavelet and was proposed in 1909 by Alfred Haar. ... Alfr d Haar (October 11, 1885 - March 16, 1933) was a Hungarian mathematician. ... George Zweig was originally trained as a particle physicist under Richard Feynman and later turned his attention to neurobiology. ... Jean Morlet is a French geophysicist who did pioneering work in the field of wavelet analysis in collaboration with Alex Grossman. ... Alex Grossman is a Croatian physicist at the University of Marseilles who did pioneering work on wavelet analysis with Jean Morlet. ... Yves Meyer (born 19 July 1939) is a French mathematician and scientist and a foremost expert on wavelets. ... StÃ©phane Mallat StÃ©phane G. Mallat made some fundamental contributions to the development of wavelet theory in the late 1980s and early 1990s. ... Ingrid Daubechies (born August 17, 1954) is a Belgian physicist and mathematician. ... Ronald Coifman is the Phillips Professor of Mathematics at Yale University. ... Mladen Victor Wickerhauser, born in Zagreb, Croatia, in 1959. ...

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: A list of wavelet related transforms: Continuous wavelet transform (CWT) Multiresolution analysis (MRA) Discrete wavelet transform (DWT) Fast wavelet transform (FWT) Complex wavelet transform Non or undecimated wavelet transform, the downsampling is omitted Newland transform, an orthonormal basis of wavelets is formed from appropriately constructed top-hat filters in frequency...

// Formulation In mathematics and signal processing, the continuous wavelet transform (CWT) of a function is a wavelet transform defined by where represents translation, represents scale and is the mother wavelet. ... In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. ... The Fast (Lifting) Wavelet Transform is a mathematical algorithm designed to turn a waveform or signal in the time domain into a sequence of coefficients based on an orthogonal basis of small finite waves, or wavelets. ... Wavelet packet decomposition (WPD) (sometimes known as just wavelet packets) is a wavelet transform where the signal is passed though more filters than the DWT. In the DWT, each level is calculated by passing the previous approximation coefficients though a high and low pass filters. ... The Stationary Wavelet Transform (SWT) (also know as the Redundant Wavelet Transform or A Trous Algorithm), is similar to the DWT except the signal is never subsampled and the filters are different for each level of decomposition. ...

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 CDF N/P or Daubechies biorthogonal wavelets)
BNC wavelets
Villasenor wavelet
Haar wavelet
Symmlet
Complex wavelet transform

Coiflet is a discrete wavelet designed by Ingrid Daubechies to be more symmetrical than the Daubechies wavelet. ... Daubechies 20 2-d wavelet (Wavelet Fn X Scaling Fn) Named after Ingrid Daubechies, the orthogonal Daubechies wavelets are a class of wavelets characterized by a maximal number of vanishing moments for some given support. ... The historically first family of biorthogonal wavelets, which was made popular by Ingrid Daubechies. ... The Haar wavelet The Haar wavelet is the first known wavelet and was proposed in 1909 by Alfred Haar. ... The complex wavelet transform is a complex-valued extension to the standard discrete wavelet transform (DWT). ...

Continuous wavelets

In numerical analysis, continuous wavelets are functions used by the continuous wavelet transform. ...

Real valued

Mexican hat wavelet
Hermitian wavelet
Hermitian hat wavelet

In mathematics and numerical analysis, the Mexican hat wavelet is the normalized, second derivative of a Gaussian function. ... Hermitian wavelets are a family of continuous wavelets, used in the continuous wavelet transform. ... The Hermitian hat wavelet is a low-oscillation, complex-valued wavelet. ...

Complex valued

The complex Mexican hat wavelet is a low-oscillation, complex-valued, wavelet for the continuous wavelet transform. ... The Morlet wavelet, named after Jean Morlet, was originally formulated by Goupillaud, Grossmann and Morlet in 1984 as a constant subtracted from a plane wave and then localised by a Gaussian: where is defined by the admissibility criterion and the normalisation constant is: The Fourier transform of the Morlet wavelet... Modified Mexican hat, Modified Morlet and Dark soliton or Darklet wavelets are derived from hyperbolic secant (sech) (bright soliton) and hyperbolic tangent (tanh) (dark soliton) pulses. ... Continuous wavelets of compact support can be built, which are related to the beta distribution. ...

References

Paul S. Addison, The Illustrated Wavelet Transform Handbook, Institute of Physics, 2002, ISBN 0-7503-0692-0
Ingrid Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, 1992, ISBN 0-89871-274-2
P. P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice Hall, 1993, ISBN 0-13-605718-7
Mladen Victor Wickerhauser, Adapted Wavelet Analysis From Theory to Software, A K Peters Ltd, 1994, ISBN 1-56881-041-5
Gerald Kaiser, A Friendly Guide to Wavelets, Birkhauser, 1994, ISBN 0-8176-3711-7
Haar A., Zur Theorie der orthogonalen Funktionensysteme, Mathematische Annalen, 69, pp 331-371, 1910.
Ramazan Gençay, Faruk Selçuk and Brandon Whitcher, An Introduction to Wavelets and Other Filtering Methods in Finance and Economics, Academic Press, 2001, ISBN 0-12-279670-5
Donald B. Percival and Andrew T. Walden, Wavelet Methods for Time Series Analysis, Cambridge University Press, 2000, ISBN 0-5216-8508-7

The Institute of Physics (IOP) is Britain and Irelands main professional body for physicists. ... Ingrid Daubechies (born August 17, 1954) is a Belgian physicist and mathematician. ...

External links

Wikimedia Commons has media related to:

Wavelet

Wavelets made Simple
Wavelet Digest
Course on Wavelets given at UC Santa Barbara, 2004
Wavelet Posting Board
The Wavelet Tutorial by Polikar (Easy to understand when you have some background with fourier transforms!)
OpenSource Wavelet C++ Code
An Introduction to Wavelets
Wavelets for Kids (PDF file) (Introductory (for very smart kids!))
Link collection about wavelets
Wavelet forums (French) Wavelet forum (English)
Gerald Kaiser's acoustic and electromagnetic wavelets
A really friendly guide to wavelets
Wavelet-based image annotation and retrieval

Categories: Numerical analysis | Signal processing | Image processing | Wavelets Image File history File links Commons-logo. ... The Wikimedia Commons (also called Wikicommons) is a repository of free content images, sound and other multimedia files. ...

Results from FactBites:

PlanetMath: wavelet (219 words)
	It is not obvious from the definition that wavelets even exist, or how to construct one; the Haar wavelet is the standard example of a wavelet, and one technique used to construct wavelets.
	Generally, wavelets are constructed from a multiresolution analysis, but they can also be generated using wavelet sets.
	This is version 6 of wavelet, born on 2004-06-24, modified 2004-06-25.

Wavelet - ExampleProblems.com (1014 words)
	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).
	All wavelet transforms may be considered to be forms of time-frequency representation and are, therefore, related to the subject of harmonic analysis.
	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.

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