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'''[[:en:Mexican hat wavelet]]'''
<nowiki>{{S|matematica}}</nowiki>
[[Image:Wavelet - Mex Hat.png|thumb|150px|Mexican hat wavelet]]
In [[matematica]] e in [[analisi numerica]], la '''wavelet cappello messicano'''
:<math>\psi(t) = {2 \over {\sqrt {3\sigma}\pi^{1 \over 4}}} \left( 1 - {t^2 \over \sigma^2} \right) e^{-t^2 \over 2\sigma^2}</math>
è la seconda [[derivata]] [[Costante normalizzante|normalizzata]] negativa di una [[funzione gaussiana]]. È un caso speciale della famiglia delle [[wavelet continue]] ([[wavelet]] usate nelle [[trasformata wavelet continua|wavelet continue trasformate]]), la famiglia delle [[wavelet hermitiane]].
Questa wavelet viene chiamata ''cappello messicano'' in quanto la sua rappresentazione su grafico ricorda un [[sombrero]], tipico cappello messicano. Tecnicamente, il suo nome è '''wavelet Ricker''', usata frequentemente per modellare dati sismici<ref>[http://www.glossary.oilfield.slb.com/Display.cfm?Term=Ricker%20wavelet Glossario Oilfield]</ref>.
In [[mathematics]] and [[numerical analysis]], the '''Mexican hat wavelet'''
:<math>\psi(t) = {2 \over {\sqrt {3\sigma}\pi^{1 \over 4}}} \left( 1 - {t^2 \over \sigma^2} \right) e^{-t^2 \over 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. It is usually only referred to as the "Mexican hat" in the Americas, due to cultural association; see "[[sombrero]]". In technical nomenclature this function is known as the '''Ricker wavelet,''' 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, because it is separable and can therefore save considerable computation time in two or more dimensions. The scale normalised 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 Mexican hat wavelet can also be approximated by [[derivative]]s of [[B-spline#Cardinal_B-spline|Cardinal B-Splines]]<ref>Brinks R: ''On the convergence of derivatives of B-splines to derivatives of the Gaussian function'', Comp. Appl. Math., 27, 1, 2008</ref>
==Note==
<references/>
<nowiki>[[Category:Numerical analysis]]</nowiki>
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