Editing Wavelet (section)

=== 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.