mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

and

numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...

, the Ricker wavelet :

\psi(t) = \frac \left(1 - \left(\frac\right)^2 \right) e^

is the negative normalized second

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...

of a

Gaussian function In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f(x) = \exp (-x^2) and with parametric extension f(x) = a \exp\left( -\frac \right) for arbitrary real constants , and non-zero . It is n ...

, i.e., up to scale and normalization, the second Hermite function. It is a special case of the family of

continuous wavelet {{Unreferenced, date=December 2009 In numerical analysis, continuous wavelets are functions used by the continuous wavelet transform. These functions are defined as analytical expressions, as functions either of time or of frequency. Most of the co ...

s (

wavelet A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...

s used in a

continuous wavelet transform Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...

) known as

Hermitian wavelet Hermitian wavelets are a family of continuous wavelets, used in the continuous wavelet transform. The n^\textrm Hermitian wavelet is defined as the n^\textrm derivative of a Gaussian distribution: \Psi_(t)=(2n)^c_He_\left(t\right)e^ where He_ ...

s. The Ricker wavelet is frequently employed to model seismic data, and as a broad spectrum source term in computational electrodynamics. It is usually only referred to as the Mexican hat wavelet in the Americas, due to taking the shape of a

sombrero A sombrero (Spanish , ) is a type of wide-brimmed Mexican men's hat used to shield the face and eyes from the sun. It usually has a high pointed crown, an extra-wide brim (broad enough to cast a shadow over the head, neck and shoulders of the we ...

when used as a 2D image processing kernel. It is also known as the Marr wavelet for David Marr. :

\psi(x,y) = \frac\left(1-\frac \left(\frac\right)\right) e^

The multidimensional generalization of this wavelet is called the

Laplacian of Gaussian In computer vision, blob detection methods are aimed at detecting regions in a digital image that differ in properties, such as brightness or color, compared to surrounding regions. Informally, a blob is a region of an image in which some proper ...

function. In practice, this wavelet is sometimes approximated by the

difference of Gaussians In imaging science, difference of Gaussians (DoG) is a feature enhancement algorithm that involves the subtraction of one Gaussian blurred version of an original image from another, less blurred version of the original. In the simple case of grays ...

(DoG) function, because the DoG is separable and 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 Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the hum ...

applications; see

and

scale space Scale-space theory is a framework for multi-scale signal representation developed by the computer vision, image processing and signal processing communities with complementary motivations from physics and biological vision. It is a formal theor ...

. The relation between this Laplacian of the Gaussian operator and the difference-of-Gaussians operator is explained in appendix A in Lindeberg (2015). The Mexican hat wavelet can also be approximated by

s of cardinal B-splines.Brinks R: ''On the convergence of derivatives of B-splines to derivatives of the Gaussian function'', Comp. Appl. Math., 27, 1, 2008

References

{{reflist Continuous wavelets

See also

References