Continuous wavelet transform: Difference between revisions

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{{Short description|Integral transform}}

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[[File:Continuous wavelet transform.svg|thumb|320px|right|Continuous [[wavelet]] transform of frequency breakdown signal. Used [[symlet]] with 5 vanishing moments.]]

In [[mathematics]], the '''continuous wavelet transform''' ('''CWT''') is a formal (i.e., non-numerical) tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of the [[wavelet]]s vary continuously.

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==Scale factor==

[[File:Continuous wavelet transform.gif|thumb|300px|right]]

The scale factor <math>a</math> either dilates or compresses a signal. When the scale factor is relatively low, the signal is more contracted which in turn results in a more detailed resulting graph. However, the drawback is that low scale factor does not last for the entire duration of the signal. On the other hand, when the scale factor is high, the signal is stretched out which means that the resulting graph will be presented in less detail. Nevertheless, it usually lasts the entire duration of the signal.

==Continuous wavelet transform properties==

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==Applications of the wavelet transform==

One of the most popular applications of wavelet transform is image compression. The advantage of using wavelet-based coding in image compression is that it provides significant improvements in picture quality at higher compression ratios over conventional techniques. Since wavelet transform has the ability to decompose complex information and patterns into elementary forms, it is commonly used in acoustics processing and pattern recognition, but it has been also proposed as an instantaneous frequency estimator.<ref>{{Cite journal|last1=Sejdic|first1=E.|last2=Djurovic|first2=I.|last3=Stankovic|first3=L.|date=August 2008|title=Quantitative Performance Analysis of Scalogram as Instantaneous Frequency Estimator|journal=IEEE Transactions on Signal Processing|volume=56|issue=8|pages=3837–3845|doi=10.1109/TSP.2008.924856|bibcode=2008ITSP...56.3837S|s2cid=16396084|issn=1053-587X}}</ref> Moreover, wavelet transforms can be applied to the following scientific research areas: edge and corner detection, partial differential equation solving, transient detection, filter design, [[electrocardiogram]] (ECG) analysis, texture analysis, business information analysis and gait analysis.<ref>[https://www.youtube.com/watch?v=DTpEVQSEBBk "Novel method for stride length estimation with body area network accelerometers"], ''IEEE BioWireless 2011'', pp. 79-8279–82</ref> Wavelet transforms can also be used in [[Electroencephalography]] (EEG) data analysis to identify epileptic spikes resulting from [[epilepsy]].<ref>{{Cite journal|last1=Iranmanesh|first1=Saam|last2=Rodriguez-Villegas|first2=Esther|author-link2=Esther Rodriguez-Villegas|year=2017|title=A 950 nW Analog-Based Data Reduction Chip for Wearable EEG Systems in Epilepsy|journal=IEEE Journal of Solid-State Circuits|volume=52|issue=9|pages=2362–2373|doi=10.1109/JSSC.2017.2720636|bibcode=2017IJSSC..52.2362I|hdl-access=free|hdl=10044/1/48764|s2cid=24852887}}</ref> Wavelet transform has been also successfully used for the interpretation of time series of landslides<ref>{{Cite journal|last1=Tomás|first1=R.|last2=Li|first2=Z.|last3=Lopez-Sanchez|first3=J. M.|last4=Liu|first4=P.|last5=Singleton|first5=A.|date=2016-06-01|title=Using wavelet tools to analyse seasonal variations from InSAR time-series data: a case study of the Huangtupo landslide|journal=Landslides|language=en|volume=13|issue=3|pages=437–450|doi=10.1007/s10346-015-0589-y|issn=1612-510X|hdl=10045/62160|s2cid=129736286|url=http://rua.ua.es/dspace/bitstream/10045/62160/5/2016_Tomas_etal_Landslides_rev.pdf|hdl-access=free}}</ref> and for calculating the changing periodicities of epidemics.<ref>{{Citation |last=von Csefalvay |first=Chris |title=Temporal dynamics of epidemics |date=2023 |url=https://linkinghub.elsevier.com/retrieve/pii/B9780323953894000165 |work=Computational Modeling of Infectious Disease |pages=217–255 |publisher=Elsevier |language=en |doi=10.1016/b978-0-32-395389-4.00016-5 |isbn=978-0-323-95389-4 |access-date=2023-02-27}}</ref>

Continuous Wavelet Transform (CWT) is very efficient in determining the damping ratio of oscillating signals (e.g. identification of damping in dynamic systems). CWT is also very resistant to the noise in the signal.<ref>Slavic, J and Simonovski, I and M. Boltezar, [http://lab.fs.uni-lj.si/ladisk/?what=abstract&ID=11 Damping identification using a continuous wavelet transform: application to real data]</ref>

==See also==

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==References==

* <ref>{{Cite journal|last1=Prasad|first1=Akhilesh|last2=Maan|first2=Jeetendrasingh|last3=Verma|first3=Sandeep Kumar|date=2021|title=Wavelet transforms associated with the index Whittaker transform|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/mma.7440|journal=Mathematical Methods in the Applied Sciences|volume=44|issue=13|pages=10734–10752|language=en|doi=10.1002/mma.7440|bibcode=2021MMAS...4410734P|s2cid=235556542|issn=1099-1476}}</ref>

*A. Grossmann & J. Morlet, 1984, Decomposition of Hardy functions into square integrable wavelets of constant shape, Soc. Int. Am. Math. (SIAM), J. Math. Analys., 15, 723-736723–736.

* Lintao Liu and Houtse Hsu (2012) "Inversion and normalization of time-frequency transform" AMIS 6 No. 1S pp. &nbsp;67S-74S.

* [[Stéphane Mallat]], "A wavelet tour of signal processing" 2nd Edition, Academic Press, 1999, {{ISBN|0-12-466606-X}}

*Ding, Jian-Jiun (2008), [http://djj.ee.ntu.edu.tw/TFW.htm Time-Frequency Analysis and Wavelet Transform], viewed 19 January 2008

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== External links ==

* {{YouTube|jnxqHcObNK4|Wavelets: a mathematical microscope}}

{{DEFAULTSORT:Continuous Wavelet Transform}}