- Wavelet
A

**wavelet**is a mathematical function used to divide a given function or continuous-time signal 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 traditionalFourier transform s for representing functions that have discontinuities and sharp peaks, and for accurately deconstructing and reconstructing finite, non-periodic and/or non-stationary signals.In formal terms, this representation is a

wavelet series representation of asquare-integrable function with respect to either a complete,orthonormal set ofbasis function s, or an overcomplete set ofFrame of a vector space (also known as aRiesz basis ), for theHilbert space of square integrable functions.Wavelet transforms are classified into

discrete wavelet transform s (DWTs) andcontinuous wavelet transform s (CWTs). Note that both DWT and CWT are continuous-time (analog) transforms. They can be used to represent continuous-time (analog) signals. CWTs operate over every possible scale and translation whereas DWTs use a specific subset of scale and translation values or representation grid. 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".**Wavelet theory**Wavelet theory is applicable to several subjects. All wavelet transforms may be considered forms of

time-frequency representation forcontinuous-time (analog) signals and so are related toharmonic analysis . Almost all practically useful discrete wavelet transforms usediscrete-time filterbank s. These filter banks are called the wavelet and scaling coefficients in wavelets nomenclature. These filterbanks may contain eitherfinite impulse response (FIR) orinfinite impulse response (IIR) filters. The wavelets forming a CWT are subject to theuncertainty principle of Fourier analysis respective sampling theory: Given a signal with some event in it, one cannot assign simultaneously an exact time and frequency response scale to that event. The product of the uncertainties of time and frequency response scale has a lower bound. Thus, in thescaleogram of a continuous wavelet transform of this signal, such an event marks an entire region in the time-scale plane, instead of just one point. This is related to Heisenberg's uncertainty principle of quantum physics and has a similar derivation. Also, discrete wavelet bases may be considered in the context of other forms of the uncertainty principle.Wavelet transforms are broadly divided into three classes: continuous, discretised and multiresolution-based.

**Continuous wavelet transforms (Continuous Shift & Scale Parameters)**In

continuous wavelet transform s, a given signal of finite energy is projected on a continuous family of frequency bands (or similar subspaces of the L^{p}function space $L^2(R)$).For instance the signal may be represented on every frequency band of the form $[f,2f]$ for all positive frequencies "f>0". Then, the original signal can be reconstructed by a suitable integration over all the resulting frequency components.The frequency bands or subspaces (sub-bands) 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 subspace of scale "a" or frequency band $[1/a,,2/a]$ is generated by the functions (sometimes called "child wavelets"):$psi\_\{a,b\}\; (t)\; =\; frac1\{sqrt\; a\; \}psi\; left(\; frac\{t\; -\; b\}\{a\}\; ight)$,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 right halfplane $R\_+\; imes\; R$.

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":$WT\_psi\{x\}(a,b)=langle\; x,psi\_\{a,b\}\; angle=int\_R\; x(t)overline\{psi\_\{a,b\}(t)\},dt$.

See a list of some

Continuous wavelets .For the analysis of the signal "x", one can assemble the wavelet coefficients into a

scaleogram of the signal.**Discrete wavelet transforms (Discrete Shift & Scale parameters)**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 $m,nin^2$. The corresponding "baby wavelets" are now given as:$psi\_\{m,n\}(t)=a^\{-m/2\}psi(a^\{-m\}t-nb)$.

A sufficient condition for the reconstruction of any signal "x" of finite energy by the formula:$x(t)=sum\_\{min^2\}sum\_\{nin^2\}langle\; x,,psi\_\{m,n\}\; anglecdotpsi\_\{m,n\}(t)$is that the functions $\{psi\_\{m,n\}:m,nin^2\}$ form a tight frame of $L^2(R)$.

**Multiresolution-based discrete wavelet transforms**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.

From the mother and father wavelets one constructs the subspaces :$V\_m=operatorname\{span\}(phi\_\{m,n\}:nin)$, where $phi\_\{m,n\}(t)=2^\{-m/2\}phi(2^\{-m\}t-n)$and :$W\_m=operatorname\{span\}(psi\_\{m,n\}:nin)$, where $psi\_\{m,n\}(t)=2^\{-m/2\}psi(2^\{-m\}t-n)$.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 thesampling 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 $[2^\{-m-1\},2^\{-m\}]$.From those inclusions and orthogonality relations follows the existence of sequences $h=\{h\_n\}\_\{nin\}$ and $g=\{g\_n\}\_\{nin\}$ that satisfy the identities:$h\_n=langlephi\_\{0,0\},,phi\_\{-1,n\}\; angle$ and $phi(t)=sqrt2\; sum\_\{nin\}\; h\_nphi(2t-n)$and :$g\_n=langlepsi\_\{0,0\},,phi\_\{-1,n\}\; angle$ and $psi(t)=sqrt2\; sum\_\{nin\}\; g\_nphi(2t-n)$.The second identity of the first pair is a refinement equation for the father wavelet $phi$. Both pairs of identities form the basis for the algorithm of the

fast wavelet transform .**Mother wavelet**For practical applications, and for efficiency reasons, one prefers 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: :$int\_\{-infty\}^\{infty\}\; |psi\; (t)|,\; dtmath>\; and$ int\_\{-infty\}^\{infty\}\; |psi\; (t)|^2\; ,\; dtmath>$>$

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 $psi$ 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.For the

discrete wavelet transform , one needs at least the condition that thewavelet series is a representation of the identity in the space $L^2(R)$. Most constructions of discreteWT make use of themultiresolution analysis , which defines the wavelet by a scaling function. This scaling function itself is solution to a functional equation.In most situations it is useful to restrict $psi$ to be a continuous function with a higher number "M" of vanishing moments, i.e. for all integer "m

int_{-infty}^{infty} t^m,psi (t), dt = 0. Some example mother wavelets are:

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 ight).

For the continuous WT, the pair "(a,b)" varies over the full half-plane $R\_+\; imesR$; 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 Transform (Continuous-Time)**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 standardFourier transform is only localized infrequency . TheShort-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 usingMultiresolution 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 . This computational advantage is not inherent to the transform, but reflects the choice of a logarithmic division of frequency, in contrast to the equally spaced frequency divisions of the FFT.**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 - 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

quadrature mirror filter 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 [

*http://perso.wanadoo.fr/polyvalens/clemens/wavelets/wavelets.html#note7*] 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 $psi\; (t)$.

For instance,

Mexican hat wavelet s can be defined by a wavelet function.See a list of a fewContinuous wavelets .**Applications of Discrete Wavelet Transform**Generally, an approximation to DWT is used for

data compression if signal is already sampled, and the CWT forsignal analysis . Thus, DWT approximation is commonly used in engineering and computer science, and the CWT in scientific research.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, includingmolecular dynamics ,ab initio calculations,astrophysics , density-matrix localisation, seismic geophysics,optics ,turbulence andquantum mechanics . This change has also occurred inimage processing , blood-pressure, heart-rate andECG analyses,DNA analysis,protein analysis,climatology , generalsignal processing ,speech recognition ,computer graphics andmultifractal analysis . Incomputer vision andimage processing , the notion ofscale-space representation and Gaussian derivative operators is regarded as a canonical multi-scale representation.One use of wavelet approximation 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" (see types ofFrame of a vector space ), 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 seewavelet compression .A related use is that of smoothing/denoising data based on wavelet coefficient thresholding, also called wavelet shrinkage. By adaptively thresholding the wavelet coefficients that correspond to undesired frequency components smoothing and/or denoising operations can be performed.

**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) [

*http://scienceworld.wolfram.com/biography/Zweig.html Zweig, George Biography on Scienceworld.wolfram.com*] , Pierre Goupillaud, Grossmann and Morlet's formulation of what is now known as the CWT (1982), Jan-Olov Strömberg's early work on discrete wavelets (1983), Daubechies' orthogonal wavelets with compact support (1988), Mallat's multiresolution framework (1989), Nathalie Delprat's time-frequency interpretation of the CWT (1991), Newland'sHarmonic wavelet transform (1993) and many others since.**Timeline*** First wavelet (

Haar wavelet ) byAlfred Haar (1909)

* Since the 1950s:George Zweig ,Jean Morlet ,Alex Grossman n

* 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)

*Lifting scheme

*Wavelet packet decomposition (WPD)

*Stationary wavelet transform (SWT)**Generalized Transforms**There are a number of generalized transforms of which the wavelet transform is a special case. For example, Joseph Segman introduced scale into the

Heisenberg group , giving rise to a continuous transform space that is a function of time, scale, and frequency. The CWT is a two-dimensional slice through the resulting 3d time-scale-frequency volume.Another example of a generalized transform is the

chirplet transform in which the CWT is also a two dimensional slice through the chirplet transform.An important application area for generalized transforms involves systems in which high frequency resolution is crucial. For example, darkfield electron optical transforms intermediate between direct and

reciprocal space have been widely used in theharmonic analysis of atom clustering, i.e. in the study ofcrystal s andcrystal defect s [*P. Hirsch, A. Howie, R. Nicholson, D. W. Pashley and M. J. Whelan (1965/1977) "Electron microscopy of thin crystals" (Butterworths, London/Krieger, Malabar FLA) ISBN 0-88275-376-2*] . Now thattransmission electron microscope s are capable of providing digital images with picometer-scale information on atomic periodicity innanostructure of all sorts, the range ofpattern recognition [*P. Fraundorf, J. Wang, E. Mandell and M. Rose (2006) Digital darkfield tableaus, "Microscopy and Microanalysis"*] and strain [**12**:S2, 1010-1011 (cf. [*http://arxiv.org/abs/cond-mat/0403017 arXiv:cond-mat/0403017*] )*M. J. Htch, E. Snoeck and R. Kilaas (1998) Quantitative measurement of displacement and strain fields from HRTEM micrographs, "Ultramicroscopy"*] /**74**:131-146.metrology [*Martin Rose (2006) "Spacing measurements of lattice fringes in HRTEM image using digital darkfield decomposition" (M.S. Thesis in Physics, U. Missouri - St. Louis)*] applications for intermediate transforms with high frequency resolution (like brushlets [*F. G. Meyer and R. R. Coifman (1997) "Applied and Computational Harmonic Analysis"*] and ridgelets [**4**:147.*A. G. Flesia, H. Hel-Or, A. Averbuch, E. J. Candes, R. R. Coifman and D. L. Donoho (2001) "Digital implementation of ridgelet packets" (Academic Press, New York).*] ) is growing rapidly.**List of wavelets****Discrete wavelets***

Beylkin (18)

*BNC wavelets

*Coiflet (6, 12, 18, 24, 30)

*Cohen-Daubechies-Feauveau wavelet (Sometimes referred to as CDF N/P or Daubechies biorthogonal wavelets)

*Daubechies wavelet (2, 4, 6, 8, 10, 12, 14, 16, 18, 20)

*Binomial-QMF

*Haar wavelet

*Mathieu wavelet

*Legendre wavelet

*Villasenor wavelet

*Symlet Continuous wavelet s**Real valued***

Beta wavelet

*Hermitian wavelet

*Hermitian hat wavelet

*Mexican hat wavelet

*Shannon wavelet **Complex valued***

Complex mexican hat wavelet

*Morlet wavelet

*Shannon wavelet

*Modified Morlet wavelet **See also***

Chirplet transform

*Curvelet

*Filter bank s

*Fractional Fourier transform

*Multiresolution analysis

*Scale space

*Short-time Fourier transform

*Ultra wideband radio- transmits wavelets.**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

* A. N. Akansu and R. A. Haddad, "Multiresolution Signal Decomposition: Transforms, Subbands, Wavelets", Academic Press, 1992, ISBN 0-12-047140-X

* 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

* Tony F. Chan and Jackie (Jianhong) Shen, "Image Processing and Analysis - Variational, PDE, Wavelet, and Stochastic Methods", Society of Applied Mathematics, ISBN 089871589X (2005)

* Stéphane Mallat, "A wavelet tour of signal processing" 2nd Edition, Academic Press, 1999, ISBN 0-12-466606-x

* Barbara Burke Hubbard, "The World According to Wavelets: The Story of a Mathematical Technique in the Making", AK Peters Ltd, 1998, ISBN 1568810725, ISBN-13 978-1568810720**Footnotes****External links*** [

*http://www.wavelet.org Wavelet Digest*]

* [*http://www.grc.nasa.gov/WWW/OptInstr/NDE_Wave_Image_ProcessorLab.html NASA Signal Processor featuring Wavelet methods*] Description of NASA Signal & Image Processing Software and Link to Download

* [*http://web.njit.edu/~ali/s1.htm 1st NJIT Symposium on Wavelets (April 30, 1990) (First Wavelets Conference in USA)*]

* [*http://web.njit.edu/~ali/NJITSYMP1990/AkansuNJIT1STWAVELETSSYMPAPRIL301990.pdf Binomial-QMF Daubechies Wavelets*]

* [*http://www.ee.ryerson.ca/~jsantarc/html/theory.html Wavelets made Simple*]

* [*http://wavelets.ens.fr/ENSEIGNEMENT/COURS/UCSB/index.html Course on Wavelets given at UC Santa Barbara, 2004*]

* [*http://users.rowan.edu/~polikar/WAVELETS/WTtutorial.html The Wavelet Tutorial by Polikar*] (Easy to understand when you have some background with fourier transforms!)

* [*http://herbert.the-little-red-haired-girl.org/en/software/wavelet/ OpenSource Wavelet C++ Code*]

* [*http://www.amara.com/IEEEwave/IEEEwavelet.html An Introduction to Wavelets*]

* [*http://www.isye.gatech.edu/~brani/wp/kidsA.pdf Wavelets for Kids (PDF file)*] (Introductory (for very smart kids!))

* [*http://www.cosy.sbg.ac.at/~uhl/wav.html Link collection about wavelets*]

* [*http://www.wavelets.com/pages/center.php Gerald Kaiser's acoustic and electromagnetic wavelets*]

* [*http://perso.wanadoo.fr/polyvalens/clemens/wavelets/wavelets.html A really friendly guide to wavelets*]

* [*http://www.alipr.com Wavelet-based image annotation and retrieval*]

* [*http://www.relisoft.com/Science/Physics/sampling.html Very basic explanation of Wavelets and how FFT relates to it*]

*Wikimedia Foundation.
2010.*