Editing Ricker wavelet

{{Short description|Wavelet proportional to the second derivative of a Gaussian}}
[[File:MexicanHatMathematica.svg|thumb|250px|Mexican hat]]

In [[mathematics]] and [[numerical analysis]], the '''Ricker wavelet''',<ref>{{Cite web| url=http://74.3.176.63/publications/recorder/1994/09sep/sep94-choice-of-wavelets.pdf | title=Ricker, Ormsby, Klauder, Butterworth - A Choice of Wavelets | accessdate=2014-12-27 | url-status=dead | archiveurl=https://web.archive.org/web/20141227215059/http://74.3.176.63/publications/recorder/1994/09sep/sep94-choice-of-wavelets.pdf | archivedate=2014-12-27}}</ref> '''Mexican hat wavelet''', or '''Marr wavelet''' (for [[David Marr (neuroscientist)|David Marr]]) <ref>{{Cite web| title=Basics of Wavelets | url=http://www2.isye.gatech.edu/~brani/isyebayes/bank/handout20.pdf | archive-url=https://web.archive.org/web/20050312150156/http://www.isye.gatech.edu:80/~brani/isyebayes/bank/handout20.pdf | archive-date=2005-03-12}}</ref><ref>{{Cite web|url=http://cxc.harvard.edu/ciao/download/doc/detect_manual/wav_theory.html#wav_theory_mh|title=13. Wavdetect Theory}}</ref>
:<math>\psi(t) = \frac{2}{\sqrt{3\sigma}\pi^{1/4}} \left(1 - \left(\frac{t}{\sigma}\right)^2 \right) e^{-\frac{t^2}{2\sigma^2}}</math>
is the negative [[normalizing constant|normalized]] second [[derivative]] of a [[Gaussian function]], i.e., up to scale and normalization, the second [[Hermite function]]. It is a special case of the family of [[continuous wavelet]]s ([[wavelet]]s used in a [[continuous wavelet transform]]) known as [[Hermitian wavelet]]s. The Ricker wavelet is frequently employed to model seismic data, and as a broad-spectrum source term in computational electrodynamics.

:<math>
\psi(x,y) = \frac{1}{\pi\sigma^4}\left(1-\frac{1}{2} \left(\frac{x^2+y^2}{\sigma^2}\right)\right) e^{-\frac{x^2+y^2}{2\sigma^2}}
</math>
[[File:Marr-wavelet2.jpg|thumb|3D view of 2D Mexican hat wavelet]]
The multidimensional generalization of this wavelet is called the [[Laplacian of Gaussian]] function. In practice, this wavelet is sometimes approximated by the [[difference of Gaussians]] (DoG) function, because the DoG is separable<ref>{{cite web|last=Fisher, Perkins, Walker and Wolfart|title=Spatial Filters - Gaussian Smoothing|url=http://homepages.inf.ed.ac.uk/rbf/HIPR2/gsmooth.htm|accessdate=23 February 2014}}</ref> and can therefore save considerable computation time in two or more dimensions.{{citation needed|date=August 2019}}{{dubious|reason=The Laplacian of Gaussian is just as easily separable as the difference of Gaussian, if the Gaussian and the Laplacian are calculated sequentially|date=August 2019}} The scale normalized Laplacian (in <math>L_1</math>-norm) is frequently used as a [[blob detection|blob detector]] and for automatic scale selection in [[computer vision]] applications; see [[Laplacian of Gaussian]] and [[scale space]]. The relation between this Laplacian of the Gaussian operator and the [[Difference of Gaussians|difference-of-Gaussians operator]] is explained in appendix A in Lindeberg (2015).<ref>{{cite journal | url=https://kth.diva-portal.org/smash/get/diva2:752787/FULLTEXT01 | doi=10.1007/s10851-014-0541-0 | title=Image Matching Using Generalized Scale-Space Interest Points | year=2015 | last1=Lindeberg | first1=Tony | journal=Journal of Mathematical Imaging and Vision | volume=52 | pages=3–36 | s2cid=254657377 | doi-access=free }}</ref> The Mexican hat wavelet can also be approximated by [[derivative]]s of [[B-spline#Cardinal B-spline|cardinal B-splines]].<ref>Brinks R: ''On the convergence of derivatives of B-splines to derivatives of the Gaussian function'', Comp. Appl. Math., 27, 1, 2008</ref>

== See also ==
* [[Morlet wavelet]]

==References==
{{reflist}}

[[Category:Continuous wavelets]]