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A '''wavelet''' is a [[wave]]-like [[oscillation]] with an [[amplitude]] that begins at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one recorded by a [[seismograph]] or [[heart monitor]]. Generally, wavelets are intentionally crafted to have specific properties that make them useful for [[signal processing]]. Using a "reverse, shift, multiply and integrate" technique called [[convolution]], wavelets can be combined with known portions of a damaged signal to extract information from the unknown portions.
[[File:Seismic Wavelet.jpg|thumb|Seismic wavelet]]
For example, a wavelet could be created to have a frequency of [[Middle C]] and a short duration of roughly a [[32nd note]]. If this wavelet were to be convolved with a signal created from the recording of a song, then the resulting signal would be useful for determining when the Middle C note was being played in the song. Mathematically, the wavelet will correlate with the signal if the unknown signal contains information of similar frequency. This concept of [[correlation]] is at the core of many practical applications of wavelet theory.

As a mathematical tool, wavelets can be used to extract information from many different kinds of data, including – but not limited to – [[audio signal]]s and images. Sets of wavelets are generally needed to analyze data fully. A set of "complementary" wavelets will decompose data without gaps or overlap so that the decomposition process is mathematically reversible. Thus, sets of complementary wavelets are useful in [[Wavelet compression|wavelet based compression]]/decompression algorithms where it is desirable to recover the original information with minimal loss.

In formal terms, this representation is a [[wavelet series]] representation of a [[square-integrable function]] with respect to either a [[Complete orthogonal system#Incomplete orthogonal sets|complete]], [[orthonormal]] set of [[basis function]]s, or an [[Complete orthogonal system#Incomplete orthogonal sets|overcomplete]] set or [[frame of a vector space]], for the [[Hilbert space]] of square integrable functions. This is accomplished through [[Coherent states in mathematical physics#The group-theoretical approach|coherent states]].

== Name ==
The word ''wavelet'' has been used for decades in digital signal processing and exploration geophysics.<ref>{{cite journal |last1=Ricker |first1=Norman |year=1953 |title=WAVELET CONTRACTION, WAVELET EXPANSION, AND THE CONTROL OF SEISMIC RESOLUTION |journal=Geophysics |volume= 18|issue= 4 |pages=769–792 |doi=10.1190/1.1437927|bibcode=1953Geop...18..769R }}</ref> The equivalent [[French language|French]] word ''ondelette'' meaning "small wave" was used by [[Jean Morlet|Morlet]] and [[Alex Grossman|Grossmann]] in the early 1980s.

== Wavelet theory ==
{{unreferenced section|date=November 2014}}
Wavelet theory is applicable to several subjects. All wavelet transforms may be considered forms of [[time-frequency representation]] for [[continuous-time]] (analog) signals and so are related to [[harmonic analysis]]. Almost all practically useful discrete wavelet transforms use [[discrete-time]] [[filterbank]]s. These filter banks are called the wavelet and scaling coefficients in wavelets nomenclature. These filterbanks may contain either [[finite impulse response]] (FIR) or [[infinite impulse response]] (IIR) filters. The wavelets forming a [[continuous wavelet transform]] (CWT) are subject to the [[Fourier uncertainty principle|uncertainty 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 the [[scaleogram]] 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. 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, discrete and multiresolution-based.

=== Continuous wavelet transforms (continuous shift and 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 [[Lp space|''L<sup>p</sup>'']] [[function space]] ''L''<sup>2</sup>('''R''') ). For instance the signal may be represented 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.

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 ψ in ''L''<sup>2</sup>('''R'''), the ''mother wavelet''. For the example of the scale one frequency band [1, 2] this function is
:<math>\psi(t)=2\,\operatorname{sinc}(2t)-\,\operatorname{sinc}(t)=\frac{\sin(2\pi t)-\sin(\pi t)}{\pi t}</math>
with the (normalized) [[sinc function]]. That, Meyer's, and two other examples of mother wavelets are:

{|
|-
| [[File:MeyerMathematica.svg|thumb|360px|[[Meyer wavelet|Meyer]]]]
|}
{|
|-
| [[File:MorletWaveletMathematica.svg|thumb|360px|[[Morlet wavelet|Morlet]]]]
|}
{|
|-
| [[File:MexicanHatMathematica.svg|thumb|360px|[[Mexican hat wavelet|Mexican hat]]]]
|}

The subspace of scale ''a'' or frequency band [1/''a'', 2/''a''] is generated by the functions (sometimes called ''child wavelets'')
:<math>\psi_{a,b} (t) = \frac1{\sqrt a }\psi \left( \frac{t - b}{a} \right),</math>
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'''<sub>+</sub> × '''R'''.

The projection of a function ''x'' onto the subspace of scale ''a'' then has the form
:<math>x_a(t)=\int_\R WT_\psi\{x\}(a,b)\cdot\psi_{a,b}(t)\,db</math>
with ''wavelet coefficients''
:<math>WT_\psi\{x\}(a,b)=\langle x,\psi_{a,b}\rangle=\int_\R x(t){\psi_{a,b}(t)}\,dt.</math>

For the analysis of the signal ''x'', one can assemble the wavelet coefficients into a [[scaleogram]] of the signal.

See a list of some [[Continuous wavelets]].

=== Discrete wavelet transforms (discrete shift and 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 [[wikt:affine|affine]] system for some real parameters ''a'' > 1, ''b'' > 0. The corresponding discrete subset of the halfplane consists of all the points (''a<sup>m</sup>'', ''na<sup>m</sup>b'') with ''m'', ''n'' in '''Z'''. The corresponding ''child wavelets'' are now given as
:<math>\psi_{m,n}(t) = \frac1{\sqrt{a^m}}\psi\left(\frac{t - nb}{a^m}\right).</math>

A sufficient condition for the reconstruction of any signal ''x'' of finite energy by the formula
:<math> x(t)=\sum_{m\in\Z}\sum_{n\in\Z}\langle x,\,\psi_{m,n}\rangle\cdot\psi_{m,n}(t)</math>
is that the functions <math>\{\psi_{m,n}:m,n\in\Z\}</math> form an [[orthonormal basis]] of ''L''<sup>2</sup>('''R''').

=== Multiresolution based discrete wavelet transforms ===
<!-- MRA:: probably Multi-resolution analysis based transforms... but
I'm not sure content matches section header... weirdness.
 -->
[[Image:Daubechies4-functions.svg|thumb|right|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. In special situations this numerical complexity can be avoided if the scaled and shifted wavelets form a [[multiresolution analysis]]. This means that there has to exist an auxiliary function, the ''father wavelet'' φ in ''L''<sup>2</sup>('''R'''), and that ''a'' is an integer. A typical choice is ''a'' = 2 and ''b'' = 1. The most famous pair of father and mother wavelets is the [[Daubechies wavelets|Daubechies]] 4-tap wavelet. Note that not every orthonormal discrete wavelet basis can be associated to a multiresolution analysis; for example, the Journe wavelet admits no multiresolution analysis.<ref>{{cite journal |last1=Larson |first1=David R. |year=2007 |title= Wavelet Analysis and Applications (See: Unitary systems and wavelet sets) |series = Appl. Numer. Harmon. Anal. |publisher=Birkhäuser |pages=143–171 }}</ref>

From the mother and father wavelets one constructs the subspaces
:<math>V_m=\operatorname{span}(\phi_{m,n}:n\in\Z),\text{ where }\phi_{m,n}(t)=2^{-m/2}\phi(2^{-m}t-n)</math>
:<math>W_m=\operatorname{span}(\psi_{m,n}:n\in\Z),\text{ where }\psi_{m,n}(t)=2^{-m/2}\psi(2^{-m}t-n).</math>
The father wavelet <math>V_{i}</math> keeps the time domain properties, while the mother wavelets <math>W_{i}</math> keeps the frequency domain properties.

From these it is required that the sequence
:<math>\{0\}\subset\dots\subset V_{1}\subset V_{0}\subset V_{-1}\subset V_{-2}\subset\dots\subset L^2(\R)</math>
forms a [[multiresolution analysis]] of ''L<sup>2</sup>'' and that the subspaces <math>\dots,W_1,W_0,W_{-1},\dots\dots</math> are the orthogonal "differences" of the above sequence, that is, ''W<sub>m</sub>'' is the orthogonal complement of ''V<sub>m</sub>'' inside the subspace ''V''<sub>''m''−1</sub>,
:<math>V_m\oplus W_m=V_{m-1}.</math>

In analogy to the [[sampling theorem]] one may conclude that the space ''V<sub>m</sub>'' with sampling distance 2<sup>''m''</sup> more or less covers the frequency baseband from 0 to 2<sup>−''m''-1</sup>. As orthogonal complement, ''W<sub>m</sub>'' roughly covers the band [2<sup>−''m''−1</sup>, 2<sup>−''m''</sup>].

From those inclusions and orthogonality relations, especially <math>V_0\oplus W_0=V_{-1}</math>, follows the existence of sequences <math>h=\{h_n\}_{n\in\Z}</math> and <math>g=\{g_n\}_{n\in\Z}</math> that satisfy the identities
:<math>g_n=\langle\phi_{0,0},\,\phi_{-1,n}\rangle</math> so that <math>\phi(t)=\sqrt2 \sum_{n\in\Z} g_n\phi(2t-n),</math> and
:<math>h_n=\langle\psi_{0,0},\,\phi_{-1,n}\rangle</math> so that <math>\psi(t)=\sqrt2 \sum_{n\in\Z} h_n\phi(2t-n).</math>
The second identity of the first pair is a [[Refinable function|refinement equation]] for the father wavelet φ. Both pairs of identities form the basis for the algorithm of the [[fast wavelet transform]].

From the multiresolution analysis derives the orthogonal decomposition of the space ''L''<sup>2</sup> as
:<math>L^2=V_{j_0}\oplus W_{j_0}\oplus W_{j_0-1}\oplus W_{j_0-2}\oplus W_{j_0-3}\oplus\dots</math>
For any signal or function <math>S\in L^2</math> this gives a representation in basis functions of the corresponding subspaces as
:<math>S = \sum_{k} c_{j_0,k}\phi_{j_0,k} + \sum_{j\le j_0}\sum_{k} d_{j,k}\psi_{j,k}</math>
where the coefficients are
:<math>c_{j_0,k} = \langle S,\phi_{j_0,k}\rangle </math> and
:<math>d_{j,k} = \langle S,\psi_{j,k}\rangle </math>.

== 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 [[Lp space|space]] <math>L^1(\R)\cap L^2(\R).</math> This is the space of [[Lebesgue integration|measurable functions]] that are absolutely and square [[integrable function|integrable]]:
:<math>\int_{-\infty}^{\infty} |\psi (t)|\, dt <\infty</math> and <math>\int_{-\infty}^{\infty} |\psi (t)|^2 \, dt <\infty.</math>

Being in this space ensures that one can formulate the conditions of zero mean and square norm one:
:<math>\int_{-\infty}^{\infty} \psi (t)\, dt = 0</math> is the condition for zero mean, and
:<math>\int_{-\infty}^{\infty} |\psi (t)|^2\, dt = 1</math> 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.

For the [[discrete wavelet transform]], one needs at least the condition that the [[wavelet series]] is a representation of the identity in the [[Lp space|space]] ''L''<sup>2</sup>('''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 a solution to a functional equation.

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''
:<math>\int_{-\infty}^{\infty} t^m\,\psi (t)\, dt = 0.</math>

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):

:<math>\psi _{a,b} (t) = {1 \over \sqrt a}\psi \left( {t - b \over a} \right).</math>

For the continuous WT, the pair (''a'',''b'') varies over the full half-plane '''R'''<sub>+</sub> × '''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).

Restriction：

(1) <math> \frac{1}{\sqrt{a}} \int_{-\infty}^\infty \varphi_{a1,b1}(t)\varphi\left(\frac{t-b}{a}\right) \, dt</math> when a1 = a and b1 = b,

(2) <math>\Psi (t)</math> has a finite time interval

== 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. In fact, the [[Fourier transform]] can be viewed as a special case of the continuous wavelet transform with the choice of the mother wavelet
<math>\psi (t) = e^{-2 \pi i t}</math>.
The main difference in general 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 similar to the wavelet transform, in that it is also time and frequency localized, but there are issues with the frequency/time resolution trade-off.

In particular, assuming a rectangular window region, one may think of the STFT as a transform with a slightly different kernel

:<math>
\psi (t) = g(t-u) e^{-2 \pi i t}
</math>

where <math>g(t-u)</math> can often be written as <math>\rm{rect}\left(\frac{t-u}{\Delta_t}\right)</math>, where <math>\Delta_t</math> and u respectively denote the length and temporal offset of the windowing function. Using [[Parseval's theorem]], one may define the wavelet's energy as
:<math>
E = \int_{-\infty}^{\infty} |\psi (t)|^2\, dt = \frac{1}{2\pi}\int_{-\infty}^{\infty} |\hat{\psi}(\omega)|^2 \, d\omega
</math>
From this, the square of the temporal support of the window offset by time u is given by

:<math>
\sigma_t^2 = \frac{1}{E}\int |t-u|^2|\psi(t)|^2 \, dt
</math>

and the square of the spectral support of the window acting on a frequency <math>\xi</math>

:<math>
\sigma_\omega^2 =\frac{1}{2\pi E} \int |\omega-\xi|^2|\hat{\psi}(\omega)|^2 \, d\omega
</math>

Multiplication with a rectangular window in the time domain corresponds to convolution with a <math>\rm{sinc(\Delta_t\omega)}</math> function in the frequency domain, resulting in spurious [[ringing artifacts]] for short/localized temporal windows. With the continuous-time Fourier Transform, <math>\Delta_t \rightarrow \infty</math> and this convolution is with a delta function in Fourier space, resulting in the true Fourier transform of the signal <math>x(t)</math>. The window function may be some other [[Apodizing|apodizing filter]], such as a [[Gaussian filter|Gaussian]]. The choice of windowing function will affect the approximation error relative to the true Fourier transform.

A given resolution cell's time-bandwidth product may not be exceeded with the STFT. All STFT basis elements maintain a uniform spectral and temporal support for all temporal shifts or offsets, thereby attaining an equal resolution in time for lower and higher frequencies. The resolution is purely determined by the sampling width.

In contrast, the wavelet transform's [[Multiresolution analysis|multiresolutional]] properties enables large temporal supports for lower frequencies while maintaining short temporal widths for higher frequencies by the scaling properties of the wavelet transform. This property extends conventional time-frequency analysis into time-scale analysis.<ref>Mallat, Stephane. "A wavelet tour of signal processing. 1998." 250-252.</ref> [[File:Time frequency atom resolution.png|thumb|STFT time-frequency atoms (left) and DWT time-scale atoms (right). The time-frequency atoms are four different basis functions used for the STFT (i.e. '''four separate Fourier transforms required'''). The time-scale atoms of the DWT achieve small temporal widths for high frequencies and good temporal widths for low frequencies with a '''single''' transform basis set.]]

The discrete wavelet transform is less computationally [[complexity|complex]], taking [[Big O notation|O(''N'')]] time as compared to O(''N''&nbsp;log&nbsp;''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 (Fast Fourier Transform) which uses the same basis functions as DFT (Discrete Fourier Transform).<ref>The Scientist and Engineer's Guide to Digital Signal Processing By Steven W. Smith, Ph.D. chapter 8 equation 8-1: http://www.dspguide.com/ch8/4.htm</ref> It is also important to note that this complexity only applies when the filter size has no relation to the signal size. A wavelet without [[compact support]] such as the [[Shannon wavelet]] would require O(''N''<sup>2</sup>). (For instance, a logarithmic Fourier Transform also exists with O(''N'') complexity, but the original signal must be sampled logarithmically in time, which is only useful for certain types of signals.<ref>http://homepages.dias.ie/~ajones/publications/28.pdf</ref>)

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

=== Scaling filter ===
An orthogonal wavelet is entirely defined by the scaling filter – a low-pass [[finite impulse response]] (FIR) filter of length 2''N'' and sum 1. In [[Biorthogonal system|biorthogonal]] wavelets, separate decomposition and reconstruction filters are defined.

For analysis with orthogonal wavelets the high pass filter is calculated as the [[quadrature mirror filter]] of the low pass, and reconstruction filters are the time reverse of the decomposition filters.

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 that 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<ref>{{cite web|url=http://www.polyvalens.com/blog/?page_id=15#7.+The+scaling+function+%5B7%5D|title=A Really Friendly Guide To Wavelets – PolyValens|author=|date=|website=www.polyvalens.com}}</ref> 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'').

For instance, [[Mexican hat wavelet]]s can be defined by a wavelet function. See a list of a few [[Continuous wavelets]].

== History ==
The development of [[wavelets]] can be linked to several separate trains of thought, starting with [[Alfréd Haar|Haar]]'s work in the early 20th century. Later work by [[Dennis Gabor]] yielded [[Gabor atom]]s (1946), which are constructed similarly to wavelets, and applied to similar purposes.

[[Wavelet compression]], a form of [[transform coding]] that uses [[wavelet transforms]] in [[data compression]], began after the development of the [[discrete cosine transform]] (DCT),<ref name="Hoffman">{{cite book |last1=Hoffman |first1=Roy |title=Data Compression in Digital Systems |date=2012 |publisher=[[Springer Science & Business Media]] |isbn=9781461560319 |page=124 |url=https://books.google.com/books?id=FOfTBwAAQBAJ |quote=Basically, wavelet coding is a variant on DCT-based transform coding that reduces or eliminates some of its limitations. (...) Another advantage is that rather than working with 8 × 8 blocks of pixels, as do JPEG and other block-based DCT techniques, wavelet coding can simultaneously compress the entire image.}}</ref> a block-based [[data compression]] algorithm first proposed by [[N. Ahmed|Nasir Ahmed]] in the early 1970s.<ref name="Ahmed">{{cite journal |last=Ahmed |first=Nasir |author-link=N. Ahmed |title=How I Came Up With the Discrete Cosine Transform |journal=[[Digital Signal Processing (journal)|Digital Signal Processing]] |date=January 1991 |volume=1 |issue=1 |pages=4–5 |doi=10.1016/1051-2004(91)90086-Z |url=https://www.scribd.com/doc/52879771/DCT-History-How-I-Came-Up-with-the-Discrete-Cosine-Transform}}</ref><ref name="Stankovic">{{cite journal |last1=Stanković |first1=Radomir S. |last2=Astola |first2=Jaakko T. |title=Reminiscences of the Early Work in DCT: Interview with K.R. Rao |journal=Reprints from the Early Days of Information Sciences |date=2012 |volume=60 |url=http://ticsp.cs.tut.fi/reports/ticsp-report-60-reprint-rao-corrected.pdf |accessdate=13 October 2019}}</ref> The introduction of the DCT led to the development of wavelet coding, a variant of DCT coding that uses wavelets instead of DCT's block-based algorithm.<ref name="Hoffman"/>

Notable contributions to wavelet theory since then can be attributed to [[George Zweig|Zweig]]’s discovery of the [[continuous wavelet transform]] (CWT) in 1975 (originally called the cochlear transform and discovered while studying the reaction of the ear to sound),<ref>http://scienceworld.wolfram.com/biography/Zweig.html
 Zweig, George Biography on Scienceworld.wolfram.com</ref> Pierre Goupillaud, [[Alex Grossman|Grossmann]] and [[Jean Morlet|Morlet]]'s formulation of what is now known as the CWT (1982), Jan-Olov Strömberg's early work on [[Strömberg wavelet|discrete wavelets]] (1983), the LeGall-Tabatabai (LGT) 5/3 wavelet developed by Didier Le Gall and Ali J. Tabatabai (1988),<ref>{{cite web |last1=Sullivan |first1=Gary |title=General characteristics and design considerations for temporal subband video coding |publisher=[[Video Coding Experts Group]] |website=[[ITU-T]] |date=8-12 December 2003 |url=https://www.itu.int/wftp3/av-arch/video-site/0312_Wai/VCEG-U06.doc |accessdate=13 September 2019}}</ref><ref>{{cite book |last1=Bovik |first1=Alan C. |title=The Essential Guide to Video Processing |date=2009 |publisher=[[Academic Press]] |isbn=9780080922508 |page=355 |url=https://books.google.com/books?id=wXmSPPB_c_0C&pg=PA355}}</ref><ref>{{cite journal |last1=Gall |first1=Didier Le |last2=Tabatabai |first2=Ali J. |title=Sub-band coding of digital images using symmetric short kernel filters and arithmetic coding techniques |journal=ICASSP-88., International Conference on Acoustics, Speech, and Signal Processing |date=1988 |pages=761–764 vol.2 |doi=10.1109/ICASSP.1988.196696}}</ref> [[Ingrid Daubechies]]' orthogonal wavelets with compact support (1988), [[Stephane Mallat|Mallat]]'s multiresolution framework (1989), [[Ali Akansu]]'s [[Binomial QMF]] (1990), Nathalie Delprat's time-frequency interpretation of the CWT (1991), Newland's [[harmonic wavelet transform]] (1993), and [[set partitioning in hierarchical trees]] (SPIHT) developed by Amir Said with William A. Pearlman in 1996.<ref name="Said">{{cite journal|last = Said|first = Amir|author2=Pearlman, William A.|title = A new fast and efficient image codec based on set partitioning in hierarchical trees|journal = IEEE Transactions on Circuits and Systems for Video Technology|volume = 6|pages = 243–250|date = June 1996|issn = 1051-8215|doi = 10.1109/76.499834|issue = 3}}</ref>

The [[JPEG 2000]] standard was developed from 1997 to 2000 by a [[Joint Photographic Experts Group]] (JPEG) committee chaired by Touradj Ebrahimi (later the JPEG president).<ref>{{cite book |last1=Taubman |first1=David |last2=Marcellin |first2=Michael |title=JPEG2000 Image Compression Fundamentals, Standards and Practice: Image Compression Fundamentals, Standards and Practice |date=2012 |publisher=[[Springer Science & Business Media]] |isbn=9781461507994 |url=https://books.google.com/books?id=y7HeBwAAQBAJ&pg=PA402}}</ref> In contrast to the DCT algorithm used by the original [[JPEG]] format, JPEG 2000 instead uses [[discrete wavelet transform]] (DWT) algorithms. It uses the [[Cohen-Daubechies-Feauveau wavelet|CDF]] 9/7 wavelet transform (developed by Ingrid Daubechies in 1992) for its [[lossy compression]] algorithm, and the LeGall-Tabatabai (LGT) 5/3 wavelet transform (developed by Didier Le Gall and Ali J. Tabatabai in 1988) for its [[lossless compression]] algorithm.<ref name="Unser">{{cite journal |last1=Unser |first1=M. |last2=Blu |first2=T. |title=Mathematical properties of the JPEG2000 wavelet filters |journal=IEEE Transactions on Image Processing |date=2003 |volume=12 |issue=9 |pages=1080–1090 |doi=10.1109/TIP.2003.812329 |url=https://pdfs.semanticscholar.org/6ed4/dece8b364416d9c390ba53df913bca7fb9a6.pdf}}</ref> [[JPEG 2000]] technology, which includes the [[Motion JPEG 2000]] extension, was selected as the [[video coding standard]] for [[digital cinema]] in 2004.<ref>{{cite book |last1=Swartz |first1=Charles S. |title=Understanding Digital Cinema: A Professional Handbook |date=2005 |publisher=[[Taylor & Francis]] |isbn=9780240806174 |page=147 |url=https://books.google.com/books?id=tYw3ehoBnjkC&pg=PA147}}</ref>

=== Timeline ===
* First wavelet ([[Haar Wavelet]]) by [[Alfréd Haar]] (1909)
* Since the 1970s: [[George Zweig]], [[Jean Morlet]], [[Alex Grossman]]n
* Since the 1980s: [[Yves Meyer]], [[Stéphane Mallat]], Didier Le Gall, Ali J. Tabatabai, [[Ingrid Daubechies]], [[Ronald Coifman]], [[Ali Akansu]], [[Victor Wickerhauser]]
* Since the 1990s: Nathalie Delprat, Newland, Amir Said, William A. Pearlman, Touradj Ebrahimi, [[JPEG 2000]]

== Wavelet transforms ==
{{main|Wavelet transform}}
A wavelet is a mathematical function used to divide a given function or [[continuous signal|continuous-time signal]] into different scale components. Usually one can assign a frequency range to each scale component. Each scale component can then be studied with a resolution that matches its scale. A wavelet transform is the representation of a function by wavelets. The wavelets are [[Scaling (geometry)|scaled]] and [[Translation (geometry)|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 transform]]s for representing functions that have discontinuities and sharp peaks, and for accurately deconstructing and reconstructing finite, non-[[Periodic function|periodic]] and/or non-[[stationary process|stationary]] signals.

Wavelet transforms are classified into [[discrete wavelet transform]]s (DWTs) and [[continuous 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.

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]] & [[Generalized lifting|Generalized Lifting Scheme]]
* [[Wavelet packet decomposition]] (WPD)
* [[Stationary wavelet transform]] (SWT)
* [[Fractional Fourier transform]] (FRFT)
* [[Fractional wavelet transform]] (FRWT)

=== Generalized transforms ===
There are a number of generalized transforms of which the wavelet transform is a special case. For example, Yosef Joseph {{Proper name|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 microscope|darkfield]] electron optical transforms intermediate between direct and [[reciprocal space]] have been widely used in the [[harmonic analysis]] of atom clustering, i.e. in the study of [[crystal]]s and [[crystal defect]]s.<ref>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}}</ref> Now that [[transmission electron microscope]]s are capable of providing digital images with picometer-scale information on atomic periodicity in [[nanostructure]] of all sorts, the range of [[pattern recognition]]<ref>P. Fraundorf, J. Wang, E. Mandell and M. Rose (2006) Digital darkfield tableaus, ''Microscopy and Microanalysis'' '''12''':S2, 1010–1011 (cf. [https://arxiv.org/abs/cond-mat/0403017 arXiv:cond-mat/0403017])</ref> and [[strain (materials science)|strain]]<ref>M. J. Hÿtch, E. Snoeck and R. Kilaas (1998) Quantitative measurement of displacement and strain fields from HRTEM micrographs, ''Ultramicroscopy'' '''74''':131-146.</ref>/[[metrology]]<ref>Martin Rose (2006) ''Spacing measurements of lattice fringes in HRTEM image using digital darkfield decomposition'' (M.S. Thesis in Physics, U. Missouri – St. Louis)</ref> applications for intermediate transforms with high frequency resolution (like brushlets<ref>F. G. Meyer and R. R. Coifman (1997) ''Applied and Computational Harmonic Analysis'' '''4''':147.</ref> and ridgelets<ref>A. G. Flesia, H. Hel-Or, A. Averbuch, [[Emmanuel Candès|E. J. Candes]], R. R. Coifman and [[David Donoho|D. L. Donoho]] (2001) ''Digital implementation of ridgelet packets'' (Academic Press, New York).</ref>) is growing rapidly.

Fractional wavelet transform (FRWT) is a generalization of the classical wavelet transform in the fractional Fourier transform domains. This transform is capable of providing the time- and fractional-domain information simultaneously and representing signals in the time-fractional-frequency plane.<ref>J. Shi, N.-T. Zhang, and X.-P. Liu, "A novel fractional wavelet transform and its applications," Sci. China Inf. Sci., vol. 55, no. 6, pp. 1270–1279, June 2012. URL: http://www.springerlink.com/content/q01np2848m388647/</ref>

== Applications of wavelet transform ==
Generally, an approximation to DWT is used for [[data compression]] if a signal is already sampled, and the CWT for [[signal analysis]].<ref>A.N. Akansu, W.A. Serdijn and I.W. Selesnick, [http://web.njit.edu/~akansu/PAPERS/ANA-IWS-WAS-ELSEVIER%20PHYSCOM%202010.pdf Emerging applications of wavelets: A review], Physical Communication, Elsevier, vol. 3, issue 1, pp. 1-18, March 2010.</ref> Thus, DWT approximation is commonly used in engineering and computer science, and the CWT in scientific research.

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 of [[Frame of a vector space|frames 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 see [[wavelet compression]].

A related use is for 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.

Wavelet transforms are also starting to be used for communication applications. Wavelet [[OFDM]] is the basic modulation scheme used in HD-PLC (a [[power line communication]]s technology developed by [[Panasonic]]), and in one of the optional modes included in the [[IEEE 1901]] standard. Wavelet OFDM can achieve deeper notches than traditional [[Fast Fourier transform|FFT]] OFDM, and wavelet OFDM does not require a guard interval (which usually represents significant overhead in FFT OFDM systems).<ref name="galli">{{cite journal |title= Recent Developments in the Standardization of Power Line Communications within the IEEE |authors= Stefano Galli, O. Logvinov |journal= IEEE Communications Magazine |date= July 2008 |volume= 46 |number= 7 |pages= 64–71 |doi= 10.1109/MCOM.2008.4557044 }} An overview of P1901 PHY/MAC proposal.</ref>

=== As a representation of a signal ===

Often, signals can be represented well as a sum of sinusoids. However, consider a non-continuous signal with an abrupt discontinuity; this signal can still be represented as a sum of sinusoids, but requires an infinite number, which is an observation known as [[Gibbs phenomenon]]. This, then, requires an infinite number of Fourier coefficients, which is not practical for many applications, such as compression. Wavelets are more useful for describing these signals with discontinuities because of their time-localized behavior (both Fourier and wavelet transforms are frequency-localized, but wavelets have an additional time-localization property). Because of this, many types of signals in practice may be non-sparse in the Fourier domain, but very sparse in the wavelet domain. This is particularly useful in signal reconstruction, especially in the recently popular field of [[compressed sensing]]. (Note that the [[short-time Fourier transform]] (STFT) is also localized in time and frequency, but there are often problems with the frequency-time resolution trade-off. Wavelets are better signal representations because of [[multiresolution analysis]].)

This motivates why wavelet transforms are now being adopted for a vast number of applications, often replacing the conventional [[Fourier Transform|Fourier transform]]. Many areas of physics have seen this paradigm shift, including [[molecular dynamics]], [[chaos theory]],<ref>{{cite journal|last1=Wotherspoon|first1=T.|last2=et.|first2=al.|title=Adaptation to the edge of chaos with random-wavelet feedback.|journal=J. Phys. Chem.|volume=113|issue=1|pages=19–22|date=2009|doi=10.1021/jp804420g|bibcode=2009JPCA..113...19W}}</ref> [[ab initio]] calculations, [[astrophysics]], [[gravitational wave]] transient data analysis,<ref>{{cite journal |collaboration=LIGO Scientific Collaboration and Virgo Collaboration |last1=Abbott |first1=Benjamin P. |title=Observing gravitational-wave transient GW150914 with minimal assumptions |journal=[[Phys. Rev. D]] |volume=93 |issue=12 |page=122004 |year=2016 |doi=10.1103/PhysRevD.93.122004 |arxiv=1602.03843 |bibcode=2016PhRvD..93l2004A |pmid=}}</ref><ref>{{cite journal |author=V Necula, S Klimenko and G Mitselmakher |title=Transient analysis with fast Wilson-Daubechies time-frequency transform |journal=Journal of Physics: Conference Series |volume=363 |issue= |page=012032 |year=2012 |doi=10.1088/1742-6596/363/1/012032 |bibcode= |pmid=}}</ref> [[Density matrix|density-matrix]] localisation, [[seismology]], [[optics]], [[turbulence]] and [[quantum mechanics]]. This change has also occurred in [[image processing]], [[Electroencephalography|EEG]], [[Electromyography|EMG]],<ref>J. Rafiee et al. Feature extraction of forearm EMG signals for prosthetics, Expert Systems with Applications 38 (2011) 4058–67.</ref> [[Electrocardiography|ECG]] analyses, [[neural oscillation|brain rhythms]], [[DNA]] analysis, [[protein]] analysis, [[climatology]], human sexual response analysis,<ref>J. Rafiee et al. Female sexual responses using signal processing techniques, The Journal of Sexual Medicine 6 (2009) 3086–96. [http://rafiee.us/files/JSM_2009.pdf (pdf)]</ref> general [[signal processing]], [[speech recognition]], acoustics, vibration signals,<ref>J. Rafiee and Peter W. Tse, Use of autocorrelation in wavelet coefficients for fault diagnosis, Mechanical Systems and Signal Processing 23 (2009) 1554–72.</ref> [[computer graphics]], [[multifractal analysis]], and [[sparse coding]]. In [[computer vision]] and [[image processing]], the notion of [[scale space]] representation and Gaussian derivative operators is regarded as a canonical multi-scale representation.

=== Wavelet denoising ===
Suppose we measure a noisy signal <math>x = s + v </math>. Assume s has a sparse representation in a certain wavelet bases, and <math>v \ \sim\ \mathcal{N}(0,\,\sigma^2I)</math>

So <math>y = W^Tx = W^Ts + W^Tv = p + z</math>.

Most elements in p are 0 or close to 0, and <math>z \ \sim\ \ \mathcal{N}(0,\,\sigma^2I)</math>

Since W is orthogonal, the estimation problem amounts to recovery of a signal in iid [[Gaussian noise]]. As p is sparse, one method is to apply a Gaussian mixture model for p.

Assume a prior <math>p \ \sim\ a\mathcal{N}(0,\,\sigma_1^2) +(1- a)\mathcal{N}(0,\,\sigma_2^2)</math>, <math>\sigma_1^2</math> is the variance of "significant" coefficients, and <math>\sigma_2^2</math> is the variance of "insignificant" coefficients.

Then <math>\tilde p = E(p/y) = \tau(y) y</math>, <math>\tau(y)</math> is called the shrinkage factor, which depends on the prior variances <math>\sigma_1^2</math> and <math>\sigma_2^2</math>. The effect of the shrinkage factor is that small coefficients are set early to 0, and large coefficients are unaltered.

Small coefficients are mostly noises, and large coefficients contain actual signal.

At last, apply the inverse wavelet transform to obtain <math> \tilde s = W \tilde p</math>

== 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, etc.)
* [[Binomial-QMF]] (Also referred to as Daubechies wavelet)
* [[Haar wavelet]]
* [[Mathieu wavelet]]
* [[Legendre wavelet]]
* [[Villasenor wavelet]]
* [[Symlet]]<ref>''Matlab Toolbox'' – URL: http://matlab.izmiran.ru/help/toolbox/wavelet/ch06_a32.html</ref>

=== Continuous wavelets ===

==== Real-valued ====
* [[Beta wavelet]]
* [[Hermitian wavelet]]
* [[Hermitian hat wavelet]]
* [[Meyer wavelet]]
* [[Mexican hat wavelet]]
* [[Poisson wavelet]]
* [[Shannon wavelet]]
* [[Spline wavelet]]
* [[Stromberg wavelet]]

==== Complex-valued ====
* [[Complex Mexican hat wavelet]]
* [[fbsp wavelet]]
* [[Morlet wavelet]]
* [[Shannon wavelet]]
* [[Modified Morlet wavelet]]

== See also ==
* [[Chirplet transform]]
* [[Curvelet]]
* [[Digital cinema]]
* [[Filter bank]]s
* [[Fractal compression]]
* [[Fractional Fourier transform]]
* [[JPEG 2000]]
* [[Multiresolution analysis]]
* [[Noiselet]]
* [[Non-separable wavelet]]
* [[Scale space]]
* [[Scaled correlation]]
* [[Shearlet]]
* [[Short-time Fourier transform]]
* [[Ultra wideband]] radio- transmits wavelets
* [[Wave packet]]
* [[Gabor wavelet#Wavelet space]]<ref>Erik Hjelmås  (1999-01-21) ''Gabor Wavelets'' URL: http://www.ansatt.hig.no/erikh/papers/scia99/node6.html</ref>
* [[Dimension reduction]]
* [[Fourier-related transforms]]
* [[Spectrogram]]
* [[Huygens–Fresnel principle]] (physical wavelets)

== References ==
=== Citations ===
{{Reflist|30em}}

=== Sources ===
* Paul S. Addison, ''The Illustrated Wavelet Transform Handbook'', [[Institute of Physics]], 2002, {{isbn|0-7503-0692-0}}
* [[Ali Akansu]] and Richard Haddad, ''Multiresolution Signal Decomposition: Transforms, Subbands, Wavelets'', Academic Press, 1992, {{isbn|0-12-047140-X}}
* B. Boashash, editor, "Time-Frequency Signal Analysis and Processing – A Comprehensive Reference", Elsevier Science, Oxford, 2003, {{isbn|0-08-044335-4}}.
* [[Tony F. Chan]] and [https://sites.google.com/view/jackieshen/ "Jackie (Jianhong) Shen"], ''Image Processing and Analysis – Variational, PDE, Wavelet, and Stochastic Methods'', Society of Applied Mathematics, {{isbn|0-89871-589-X}} (2005)
* [[Ingrid Daubechies]], ''Ten Lectures on Wavelets'', Society for Industrial and Applied Mathematics, 1992, {{isbn|0-89871-274-2}}
* 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}}
* Haar A., ''Zur Theorie der orthogonalen Funktionensysteme'', Mathematische Annalen, '''69''', pp 331–371, 1910.
* Barbara Burke Hubbard, "The World According to Wavelets: The Story of a Mathematical Technique in the Making", AK Peters Ltd, 1998, {{isbn|1-56881-072-5}}, {{isbn|978-1-56881-072-0}}
* Gerald Kaiser, ''A Friendly Guide to Wavelets'', Birkhauser, 1994, {{isbn|0-8176-3711-7}}
* [[Stéphane Mallat]], "A wavelet tour of signal processing" 2nd Edition, Academic Press, 1999, {{isbn|0-12-466606-X}}
* Donald B. Percival and Andrew T. Walden, ''Wavelet Methods for Time Series Analysis'', Cambridge University Press, 2000, {{isbn|0-521-68508-7}}
*{{Citation | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press |  location=New York | isbn=978-0-521-88068-8 | chapter=Section 13.10. Wavelet Transforms | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=699}}
* [[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}}
* Martin Vetterli and Jelena Kovačević, "Wavelets and Subband Coding", Prentice Hall, 1995, {{isbn|0-13-097080-8}}

== External links ==
{{extlinks|section|date=July 2016}}
{{Wiktionary|wavelet}}
{{commons|Wavelet|Wavelet}}
* [http://www.wavelet.org Wavelet Digest]
* [http://wavelets.org/software.php Wavelets: Software] – a list of useful wavelet transform frameworks, libraries, and other software
* {{springer|title=Wavelet analysis|id=p/w097160}}
* [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-math.mit.edu/~gs/papers/amsci.pdf Wavelets] by Gilbert Strang, American Scientist 82 (1994) 250–255. (A very short and excellent introduction)
* [http://wavelets.ens.fr/ENSEIGNEMENT/COURS/UCSB/index.html Course on Wavelets given at UC Santa Barbara, 2004]
* [http://www.isye.gatech.edu/~brani/wp/kidsA.pdf Wavelets for Kids (PDF file)] (Introductory (for very smart kids!))
* [http://www.relisoft.com/Science/Physics/sampling.html Very basic explanation of Wavelets and how FFT relates to it]
* [http://www.laurent-duval.eu/siva-wits-where-is-the-starlet.html WITS: Where Is The Starlet?] A dictionary of tens of wavelets and wavelet-related terms ending in -let, from activelets to x-lets through bandlets, contourlets, curvelets, noiselets, wedgelets.
* [http://bigwww.epfl.ch/publications/blu0001.pdf The Fractional Spline Wavelet Transform] describes a [[fractional wavelet transform]] based on fractional b-Splines.
* [https://dx.doi.org/10.1016/j.sigpro.2011.04.025 A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity] provides a tutorial on two-dimensional oriented wavelets and related geometric multiscale transforms.
* [http://tx.technion.ac.il/~rc/SignalDenoisingUsingWavelets_RamiCohen.pdf Signal Denoising using Wavelets]
* [https://www.scribd.com/document/370574131/Concise-Introduction-to-Wavelets Concise Introduction to Wavelets by René Puchinger]

{{Statistics|analysis}}

[[Category:Wavelets| ]]
[[Category:Time–frequency analysis]]
[[Category:Signal processing]]