Content deleted Content added
Luminia2001 (talk | contribs) m →General overview: There are examples of L¹&&L² functions that do not decay at infinity; taking such an example and making the substitution ω->1/ω now clearly gives a counterexample to "an admissible wavelet must have its Fourier transform, evaulated at 0, exactly zero". Tags: Mobile edit Mobile app edit Android app edit |
m Open access bot: url-access=subscription updated in citation with #oabot. |
||
| (11 intermediate revisions by 10 users not shown) | |||
Line 1:
{{Short description|Integral transform}}
{{more citations needed|date=June 2012}}
{{Use dmy dates|date=June 2023}}
[[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.
==Definition==
The continuous wavelet transform of a function <math>x(t)</math> at a scale
:<math display="block">X_w(a,b)=\frac{1}{|a|^{1/2}} \int_{-\infty}^\infty x(t)\overline\psi\left(\frac{t-b}{a}\right)\,
where <math>\psi(t)</math> is a continuous function in both the time ___domain and the frequency ___domain called the mother wavelet and the overline represents operation of [[complex conjugate]]. The main purpose of the mother wavelet is to provide a source function to generate the daughter wavelets which are simply the translated and scaled versions of the mother wavelet. To recover the original signal <math>x(t)</math>, the first inverse continuous wavelet transform can be exploited.
:<math>x(t)=C_\psi^{-1}\int_{0}^{\infty}\int_{-\infty}^{\infty} X_w(a,b)\frac{1}{|a|^{1/2}}\tilde\psi\left(\frac{t-b}{a}\right)\,
<math>\tilde\psi(t)</math> is the [[Dual wavelet|dual function]] of <math>\psi(t)</math> and
:<math>C_\psi=\int_{-\infty}^{\infty}\frac{\overline\hat{\psi}(\omega)\hat{\tilde\psi}(\omega)}{|\omega|}\, \mathrm{d}\omega</math>
is admissible constant, where hat means Fourier transform operator. Sometimes, <math>\tilde\psi(t)=\psi(t)</math>, then the admissible constant becomes
:<math>C_\psi = \int_{-\infty}^{+\infty}
\frac{\left| \hat{\psi}(\omega) \right|^2}{\left| \omega \right|} \, \mathrm{d}\omega
</math>
Traditionally, this constant is called wavelet admissible constant. A wavelet whose admissible constant satisfies
:<math>0<C_\psi <\infty</math>
is called an admissible wavelet. To recover the original signal <math>x(t)</math>, the second inverse continuous wavelet transform can be exploited.
:<math>x(t)=\frac{1}{2\pi\overline\hat{\psi}(1)}\int_{0}^{\infty}\int_{-\infty}^{\infty} \frac{1}{a^2}X_w(a,b)\exp\left(i\frac{t-b}{a}\right)\,
This inverse transform suggests that a wavelet should be defined as
:<math>\psi(t)=w(t)\exp(it) </math>
Line 40 ⟶ 37:
==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. 79–82</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|bibcode=2016Lands..13..437T |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 land subsidence,<ref>{{Cite journal |last1=Tomás |first1=Roberto |last2=Pastor |first2=José Luis |last3=Béjar-Pizarro |first3=Marta |last4=Bonì |first4=Roberta |last5=Ezquerro |first5=Pablo |last6=Fernández-Merodo |first6=José Antonio |last7=Guardiola-Albert |first7=Carolina |last8=Herrera |first8=Gerardo |last9=Meisina |first9=Claudia |last10=Teatini |first10=Pietro |last11=Zucca |first11=Francesco |last12=Zoccarato |first12=Claudia |last13=Franceschini |first13=Andrea |date=2020-04-22 |title=Wavelet analysis of land subsidence time-series: Madrid Tertiary aquifer case study |url=https://piahs.copernicus.org/articles/382/353/2020/ |journal=Proceedings of the International Association of Hydrological Sciences |language=en |volume=382 |pages=353–359 |doi=10.5194/piahs-382-353-2020 |doi-access=free |bibcode=2020PIAHS.382..353T |issn=2199-899X|hdl=11577/3338112 |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|url-access=subscription }}</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>
Line 48 ⟶ 45:
* [[S transform]]
* [[Time-frequency analysis]]
* [[Cauchy wavelet]]
==References==
{{Reflist}}▼
=== Further reading===
*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, 723–736.
* Lintao Liu and Houtse Hsu (2012) "Inversion and normalization of time-frequency transform" AMIS 6 No. 1S pp. 67S-74S.
Line 60 ⟶ 59:
*Valens, Clemens (2004), [http://www.polyvalens.com/blog/wavelets/ A Really Friendly Guide to Wavelets], viewed 18 September 2018]
*[http://reference.wolfram.com/mathematica/ref/ContinuousWaveletTransform.html Mathematica Continuous Wavelet Transform]
▲{{Reflist}}
== External links ==
| |||