Ricker wavelet
In mathematics and numerical analysis, the Ricker wavelet,<ref>Template:Cite web</ref> Mexican hat wavelet, or Marr wavelet (for David Marr) <ref>Template:Cite web</ref><ref>Template:Cite web</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 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.
- <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>
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>Template:Cite web</ref> It can therefore save considerable computation time in two or more dimensions.Template:Citation needed{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= Template:Fix }} The scale-normalized Laplacian (in <math>L_1</math>-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).<ref>Template:Cite journal</ref> Derivatives of 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>