Editing Wavelet (section)

== Wavelet transforms ==
{{main|Wavelet transform}}
A wavelet is a mathematical function used to divide a given function or [[continuous signal|continuous-time signal]] into different scale components. Usually one can assign a frequency range to each scale component. Each scale component can then be studied with a resolution that matches its scale. A wavelet transform is the representation of a function by wavelets. The wavelets are [[Scaling (geometry)|scaled]] and [[Translation (geometry)|translated]] copies (known as "daughter wavelets") of a finite-length or fast-decaying oscillating waveform (known as the "mother wavelet"). Wavelet transforms have advantages over traditional [[Fourier transform]]s for representing functions that have discontinuities and sharp peaks, and for accurately deconstructing and reconstructing finite, non-[[Periodic function|periodic]] and/or non-[[stationary process|stationary]] signals.

Wavelet transforms are classified into [[discrete wavelet transform]]s (DWTs) and [[continuous wavelet transform]]s (CWTs). Note that both DWT and CWT are continuous-time (analog) transforms. They can be used to represent continuous-time (analog) signals. CWTs operate over every possible scale and translation whereas DWTs use a specific subset of scale and translation values or representation grid.

There are a large number of wavelet transforms each suitable for different applications. For a full list see [[list of wavelet-related transforms]] but the common ones are listed below:
* [[Continuous wavelet transform]] (CWT)
* [[Discrete wavelet transform]] (DWT)
* [[Fast wavelet transform]] (FWT)
* [[Lifting scheme]] and [[Generalized lifting|generalized lifting scheme]]
* [[Wavelet packet decomposition]] (WPD)
* [[Stationary wavelet transform]] (SWT)
* [[Fractional Fourier transform]] (FRFT)
* [[Fractional wavelet transform]] (FRWT)

=== Generalized transforms ===
There are a number of generalized transforms of which the wavelet transform is a special case. For example, Yosef Joseph {{Proper name|Segman}} introduced scale into the [[Heisenberg group]], giving rise to a continuous transform space that is a function of time, scale, and frequency. The CWT is a two-dimensional slice through the resulting 3d time-scale-frequency volume.

Another example of a generalized transform is the [[chirplet transform]] in which the CWT is also a two dimensional slice through the chirplet transform.

An important application area for generalized transforms involves systems in which high frequency resolution is crucial. For example, [[darkfield microscope|darkfield]] electron optical transforms intermediate between direct and [[reciprocal space]] have been widely used in the [[harmonic analysis]] of atom clustering, i.e. in the study of [[crystal]]s and [[crystal defect]]s.<ref>P. Hirsch, A. Howie, R. Nicholson, D. W. Pashley and M. J. Whelan (1965/1977) ''Electron microscopy of thin crystals'' (Butterworths, London/Krieger, Malabar FLA) {{isbn|0-88275-376-2}}</ref> Now that [[transmission electron microscope]]s are capable of providing digital images with picometer-scale information on atomic periodicity in [[nanostructure]] of all sorts, the range of [[pattern recognition]]<ref>P. Fraundorf, J. Wang, E. Mandell and M. Rose (2006) Digital darkfield tableaus, ''Microscopy and Microanalysis'' '''12''':S2, 1010–1011 (cf. [https://arxiv.org/abs/cond-mat/0403017 arXiv:cond-mat/0403017])</ref> and [[strain (materials science)|strain]]<ref>{{cite journal | last1 = Hÿtch | first1 = M. J. | last2 = Snoeck | first2 = E. | last3 = Kilaas | first3 = R. | year = 1998 | title = Quantitative measurement of displacement and strain fields from HRTEM micrographs | journal = Ultramicroscopy | volume = 74 | issue = 3| pages = 131–146 | doi=10.1016/s0304-3991(98)00035-7}}</ref>/[[metrology]]<ref>Martin Rose (2006) ''Spacing measurements of lattice fringes in HRTEM image using digital darkfield decomposition'' (M.S. Thesis in Physics, U. Missouri – St. Louis)</ref> applications for intermediate transforms with high frequency resolution (like brushlets<ref>F. G. Meyer and R. R. Coifman (1997) ''[[Applied and Computational Harmonic Analysis]]'' '''4''':147.</ref> and ridgelets<ref>A. G. Flesia, [[Hagit Hel-Or|H. Hel-Or]], A. Averbuch, [[Emmanuel Candès|E. J. Candes]], R. R. Coifman and [[David Donoho|D. L. Donoho]] (2001) ''Digital implementation of ridgelet packets'' (Academic Press, New York).</ref>) is growing rapidly.

Fractional wavelet transform (FRWT) is a generalization of the classical wavelet transform in the fractional Fourier transform domains. This transform is capable of providing the time- and fractional-domain information simultaneously and representing signals in the time-fractional-frequency plane.<ref>{{cite journal | last1 = Shi | first1 = J. | last2 = Zhang | first2 = N.-T. | last3 = Liu | first3 = X.-P. | year = 2011| title = A novel fractional wavelet transform and its applications | journal = Sci. China Inf. Sci. | volume = 55 | issue = 6| pages = 1270–1279 | doi=10.1007/s11432-011-4320-x| s2cid = 255201598 | doi-access =  }}</ref>