Editing Wavelet (section)

== Wavelet theory ==
{{More citations needed|date=May 2025}}
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]].<ref>{{Cite web |title=Continuous wavelet transform - Knowledge and References |url=https://taylorandfrancis.com/knowledge/Engineering_and_technology/Engineering_support_and_special_topics/Continuous_wavelet_transform/#:~:text=The%20wavelet%20theory%20applies%20to,banks%20of%20filters%20in%20time. |access-date=2024-11-27 |website=Taylor & Francis |language=en-US}}</ref> Discrete wavelet transform (continuous in time) of a [[discrete-time]] (sampled) signal by using [[discrete-time]] [[filterbank]]s of dyadic (octave band) configuration is a wavelet approximation to that signal. The coefficients of such a filter bank are called the shift 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:<ref>{{Cite web |title=Continuous wavelet transform - Knowledge and References |url=https://taylorandfrancis.com/knowledge/Engineering_and_technology/Engineering_support_and_special_topics/Continuous_wavelet_transform/ |access-date=2024-11-27 |website=Taylor & Francis |language=en-US}}</ref> 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.<ref>Meyer, Yves (1992), Wavelets and Operators, Cambridge, UK: Cambridge University Press, {{ISBN|0-521-42000-8}}</ref><ref>Chui, Charles K. (1992), An Introduction to Wavelets, San Diego, CA: Academic Press, {{ISBN|0-12-174584-8}}</ref><ref>Daubechies, Ingrid. (1992), Ten Lectures on Wavelets, SIAM, {{ISBN|978-0-89871-274-2}}</ref><ref>Akansu, Ali N.; Haddad, Richard A. (1992), Multiresolution Signal Decomposition: Transforms, Subbands, and Wavelets, Boston, MA: Academic Press, {{ISBN|978-0-12-047141-6}}</ref>

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 display="block">\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:

{|
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| [[File:MeyerMathematica.svg|thumb|360px|[[Meyer wavelet|Meyer]]]]
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{|
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| [[File:MorletWaveletMathematica.svg|thumb|360px|[[Morlet wavelet|Morlet]]]]
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{|
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| [[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 display="block">\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 display="block">x_a(t)=\int_\R WT_\psi\{x\}(a,b)\cdot\psi_{a,b}(t)\,db</math>
with ''wavelet coefficients''
<math display="block">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, continuous in time) ===
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 transformation|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>'', ''nb a<sup>m</sup>'') with ''m'', ''n'' in '''Z'''. The corresponding ''child wavelets'' are now given as
<math display="block">\psi_{m,n}(t) = \frac1{\sqrt{a^m}}\psi\left(\frac{t - nba^m}{a^m}\right).</math>

A sufficient condition for the reconstruction of any signal ''x'' of finite energy by the formula
<math display="block"> 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 (continuous in time)===
<!-- MRA:: probably Multi-resolution analysis based transforms... but
I'm not sure content matches section header... weirdness.
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[[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>{{citation |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 display="block">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 display="block">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 display="block">\{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</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 display="block">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 1/2<sup>''m''-1</sup>. As orthogonal complement, ''W<sub>m</sub>'' roughly covers the band [1/2<sup>''m''−1</sup>, 1/2<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 display="block">g_n=\langle\phi_{0,0},\,\phi_{-1,n}\rangle</math> so that <math display="inline">\phi(t)=\sqrt2 \sum_{n\in\Z} g_n\phi(2t-n),</math> and
<math display="block">h_n=\langle\psi_{0,0},\,\phi_{-1,n}\rangle</math> so that <math display="inline">\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 display="block">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 \cdots</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 display="block">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 display="block">c_{j_0,k} = \langle S,\phi_{j_0,k}\rangle </math> and
<math display="block">d_{j,k} = \langle S,\psi_{j,k}\rangle. </math>

===Time-causal wavelets===

For processing temporal signals in real time, it is essential that the wavelet filters do not access signal values from the future as well as that minimal temporal latencies can be obtained. Time-causal wavelets representations have been developed by Szu et al <ref>{{cite journal | doi=10.1117/12.59911 | title=Causal analytical wavelet transform | year=1992 | last1=Szu | first1=Harold H. | last2=Telfer | first2=Brian A. | last3=Lohmann | first3=Adolf W. | journal=Optical Engineering | volume=31 | issue=9 | page=1825 | bibcode=1992OptEn..31.1825S }}</ref> and Lindeberg,<ref>{{cite journal |last1=Lindeberg |first1=T. |title=A time-causal and time-recursive scale-covariant scale-space representation of temporal signals and past time |journal=Biological Cybernetics |date=23 January 2023 |volume=117 |issue=1–2 |pages=21–59 |doi=10.1007/s00422-022-00953-6|pmid=36689001 |pmc=10160219 |doi-access=free }}</ref> with the latter method also involving a memory-efficient time-recursive implementation.