Ricker wavelet

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title: "Ricker wavelet" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["continuous-wavelets"] description: "Wavelet proportional to the second derivative of a Gaussian" topic_path: "general/continuous-wavelets" source: "https://en.wikipedia.org/wiki/Ricker_wavelet" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Wavelet proportional to the second derivative of a Gaussian ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/0/08/MexicanHatMathematica.svg" caption="Mexican hat"] ::

In mathematics and numerical analysis, the Ricker wavelet, Mexican hat wavelet, or Marr wavelet (for David Marr) :\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 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.

: \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}} ::figure[src="https://upload.wikimedia.org/wikipedia/commons/7/7e/Marr-wavelet2.jpg" caption="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. It can therefore save considerable computation time in two or more dimensions. The scale-normalized Laplacian (in 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.

References

References

1. "Ricker, Ormsby, Klauder, Butterworth - A Choice of Wavelets".
2. "Basics of Wavelets".
3. "13. Wavdetect Theory".
4. Fisher, Perkins, Walker and Wolfart. "Spatial Filters - Gaussian Smoothing".
5. (2015). "Image Matching Using Generalized Scale-Space Interest Points". Journal of Mathematical Imaging and Vision.
6. 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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