In mathematics and numerical analysis, the Ricker wavelet, Mexican hat wavelet, or Marr wavelet (for David Marr)
ψ ( t ) = 2 3 σ π 1 / 4 ( 1 − ( t σ ) 2 ) e − t 2 2 σ 2 {\displaystyle \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}}}}}
is the negative normalized ( ∫ R ψ 2 = 1 ) {\displaystyle \left(\int _{\mathbb {R} }\psi ^{2}=1\right)} 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 2-dimensional generalization
ψ ( x , y ) = 1 π σ 4 ( 1 − 1 2 ( x 2 + y 2 σ 2 ) ) e − x 2 + y 2 2 σ 2 {\displaystyle \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}}}}}
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. It can therefore save considerable computation time in two or more dimensions. The scale-normalized Laplacian (in L 1 {\displaystyle L_{1}} -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). Derivatives of cardinal B-splines can also approximate the Mexican hat wavelet.
See also Morlet wavelet
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