Editing Wavelet (section)

== Applications ==
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><ref>Tomás, R., Li, Z., Lopez-Sanchez, J.M., Liu, P. & Singleton, A. 2016. Using wavelet tools to analyse seasonal variations from InSAR time-series data: a case study of the Huangtupo landslide. Landslides, 13, 437-450, doi: 10.1007/s10346-015-0589-y.
</ref> Thus, DWT approximation is commonly used in engineering and computer science,<ref>{{Cite journal |last1=Lyakhov |first1=Pavel |last2=Semyonova |first2=Nataliya |last3=Nagornov |first3=Nikolay |last4=Bergerman |first4=Maxim |last5=Abdulsalyamova |first5=Albina |date=2023-11-14 |title=High-Speed Wavelet Image Processing Using the Winograd Method with Downsampling |journal=Mathematics |language=en |volume=11 |issue=22 |pages=4644 |doi=10.3390/math11224644 |issn=2227-7390 |quote="Wavelets are actively used to solve a wide range of image processing problems in various fields of science and technology, e.g., image denoising, reconstruction, analysis, and video analysis and processing. Wavelet processing methods are based on the discrete wavelet transform using 1D digital filtering." |doi-access=free }}</ref> and the CWT in scientific research.<ref>{{Cite journal |last1=Dong |first1=Liang |last2=Zhang |first2=Shaohua |last3=Gan |first3=Tiansiyu |last4=Qiu |first4=Yan |last5=Song |first5=Qinfeng |last6=Zhao |first6=Yongtao |date=2023-12-01 |title=Frequency characteristics analysis of pipe-to-soil potential under metro stray current interference using continuous wavelet transform method |url=https://www.sciencedirect.com/science/article/pii/S0950061823031707 |journal=Construction and Building Materials |volume=407 |pages=133453 |doi=10.1016/j.conbuildmat.2023.133453 |s2cid=263317973 |issn=0950-0618}}</ref>

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 |author1= Stefano Galli |author2=O. Logvinov |journal= IEEE Communications Magazine |date= July 2008 |volume= 46 |number= 7 |pages= 64–71 |doi= 10.1109/MCOM.2008.4557044 |s2cid= 2650873 }} 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|pmid=19072712}}</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 |s2cid=119313566 }}</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 |page=012032 |year=2012 |issue=1 |doi=10.1088/1742-6596/363/1/012032 |bibcode=2012JPhCS.363a2032N |doi-access=free }}</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>{{cite journal | last1 = Rafiee | first1 = J. | last2 = Tse | first2 = Peter W. | year = 2009 | title = Use of autocorrelation in wavelet coefficients for fault diagnosis | journal = Mechanical Systems and Signal Processing | volume = 23 | issue = 5| pages = 1554–72 | doi=10.1016/j.ymssp.2009.02.008| bibcode = 2009MSSP...23.1554R }}</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 ===

[[File:Wavelet denoising.svg|thumb|Signal denoising by wavelet transform thresholding]]

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

Let the wavelet transform of <math>x</math> be <math>y = W^T x = W^T s + W^T v = p + z</math>, where <math>p = W^T s</math> is the wavelet transform of the signal component and <math>z = W^T v</math> is the wavelet transform of the noise component.

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

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

Assume a prior <math>p \ \sim\ a\mathcal{N}(0,\,\sigma_1^2) +(1- a)\mathcal{N}(0,\,\sigma_2^2)</math>, where <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>. By setting coefficients that fall below a shrinkage threshold to zero, once the inverse transform is applied, an expectedly small amount of signal is lost due to the sparsity assumption. The larger coefficients are expected to primarily represent signal due to sparsity, and statistically very little of the signal, albeit the majority of the noise, is expected to be represented in such lower magnitude coefficients... therefore the zeroing-out operation is expected to remove most of the noise and not much signal. Typically, the above-threshold coefficients are not modified during this process. Some algorithms for wavelet-based denoising may attenuate larger coefficients as well, based on a statistical estimate of the amount of noise expected to be removed by such an attenuation.

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

=== Multiscale climate network ===

Agarwal et al. proposed wavelet based advanced linear <ref name="AgarwalMaheswaranMarwan2018">{{cite journal | last1 = Agarwal | first1 = Ankit | last2 = Maheswaran | first2 = Rathinasamy | last3 = Marwan | first3 = Norbert | last4 = Caesar | first4 = Levke | last5 = Kurths | first5 = Jürgen | title = Wavelet-based multiscale similarity measure for complex networks | journal = The European Physical Journal B | date = November 2018 | volume = 91 | issue = 11 | page = 296 | issn = 1434-6028 | eissn = 1434-6036 | doi = 10.1140/epjb/e2018-90460-6 | pmid = | bibcode = 2018EPJB...91..296A | s2cid = 125557123 | url = http://mural.maynoothuniversity.ie/13175/1/CL_geography_wavelet-based.pdf}}</ref> and nonlinear <ref name="AgarwalMarwanRathinasamy2017">{{cite journal | last1 = Agarwal | first1 = Ankit | last2 = Marwan | first2 = Norbert | last3 = Rathinasamy | first3 = Maheswaran | last4 = Merz | first4 = Bruno | last5 = Kurths | first5 = Jürgen | title = Multi-scale event synchronization analysis for unravelling climate processes: a wavelet-based approach | journal = Nonlinear Processes in Geophysics | date = 13 October 2017 | volume = 24 | issue = 4 | pages = 599–611 | eissn = 1607-7946 | doi = 10.5194/npg-24-599-2017 | pmid = | bibcode = 2017NPGeo..24..599A | s2cid = 28114574 | url = | doi-access = free }}</ref> methods to construct and investigate [[Climate as complex networks]] at different timescales. Climate networks constructed using [[Sea surface temperature|SST]] datasets at different timescale averred that wavelet based multi-scale analysis of climatic processes holds the promise of better understanding the system dynamics that may be missed when processes are analyzed at one timescale only
<ref name="AgarwalCaesarMarwan2019">{{cite journal | last1 = Agarwal | first1 = Ankit | last2 = Caesar | first2 = Levke | last3 = Marwan | first3 = Norbert | last4 = Maheswaran | first4 = Rathinasamy | last5 = Merz | first5 = Bruno | last6 = Kurths | first6 = Jürgen | title = Network-based identification and characterization of teleconnections on different scales | journal = Scientific Reports | date = 19 June 2019 | volume = 9 | issue = 1 | page = 8808 | eissn = 2045-2322 | doi = 10.1038/s41598-019-45423-5 | pmid = 31217490 | pmc = 6584743 | bibcode = 2019NatSR...9.8808A | url = }}</ref>