mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

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, Mexican hat wavelet, or Marr wavelet (for David Marr) :

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

is the negative

normalized Normalization or normalisation refers to a process that makes something more normal or regular. Science * Normalization process theory, a sociological theory of the implementation of new technologies or innovations * Normalization model, used in ...

second

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

of a

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

, i.e., up to scale and normalization, the second

Hermite function In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well as ...

. It is a special case of the family of

continuous wavelet 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 continuous wavelets are used for both ...

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 n ...

s used in a

continuous wavelet transform In mathematics, the continuous wavelet transform (CWT) is a formal (i.e., non-numerical) tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of the wavelets vary continuously. Definition ...

) known as

Hermitian wavelet Hermitian wavelets are a family of discrete and continuous wavelets used in the constant and discrete Hermite wavelet transforms. The n^\textrm Hermitian wavelet is defined as the normalized n^\textrm derivative of a Gaussian distribution for ea ...

s. The Ricker wavelet is frequently employed to model seismic data, and as a broad-spectrum source term in computational electrodynamics. :

\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 and image processing, 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 a ...

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 g ...

(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 tasks include methods for image sensor, acquiring, Image processing, processing, Image analysis, analyzing, and understanding digital images, and extraction of high-dimensional data from the real world in order to produce numerical ...

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 the ...

. 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