Mexican hat wavelet

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Mexican hat wavelet
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Mexican hat wavelet

In mathematics and numerical analysis, the Mexican hat wavelet

\psi(t) = \left( {1 \over {\sqrt {2\pi}\sigma^3}} \right) \left( \left( 1 - {t^2 \over \sigma^2} \right) e^{-t^2 \over 2\sigma^2}\right)

is the normalized second derivative of a Gaussian function. It is a special case of the family of continuous wavelets (wavelets used in a continuous wavelet transform) known as Hermitian wavelets.

The Mexican hat wavelet is often called the Ricker Wavelet in Geophysics, where it is frequently employed to model seismic data.

The hyperdimensional generalization of this wavelet is called the Laplacian of Gaussian function. In practice, this wavelet is sometimes approximated by the Difference of Gaussians function, with the argument that differences of Gaussians are claimed to be easier to compute. With a proper discretization, however, this argument can be questioned. The scale normalised Laplacian (in L1-norm) is frequently used as a blob detector and for automatic scale selection in computer vision applications; see scale-space.

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