Ricker wavelet: Difference between revisions
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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> | 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> | :<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 | 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> | :<math> | ||
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</math> | </math> | ||
[[File:Marr-wavelet2.jpg|thumb|3D view of 2D Mexican hat wavelet]] | [[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> | 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> It 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> [[Derivative]]s of [[B-spline#Cardinal B-spline|cardinal B-splines]] can also approximate the Mexican hat wavelet.<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 == | == See also == | ||
Latest revision as of 10:21, 20 August 2025
In mathematics and numerical analysis, the Ricker wavelet,[1] Mexican hat wavelet, or Marr wavelet (for David Marr) [2][3]
is the negative 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 wavelets (wavelets used in a continuous wavelet transform) known as Hermitian wavelets. The Ricker wavelet is frequently employed to model seismic data and as a broad-spectrum source term in computational electrodynamics.
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.[4] It can therefore save considerable computation time in two or more dimensions.Script error: No such module "Unsubst".Script error: No such module "Unsubst". The scale-normalized Laplacian (in -norm) is frequently used as a 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 operator is explained in appendix A in Lindeberg (2015).[5] Derivatives of cardinal B-splines can also approximate the Mexican hat wavelet.[6]
See also
References
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- ↑ Brinks R: On the convergence of derivatives of B-splines to derivatives of the Gaussian function, Comp. Appl. Math., 27, 1, 2008
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