numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...
, continuous
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 are functions used by the
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 ...
. These functions are defined as
analytical expression
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th ro ...
s, as functions either of time or of frequency.
Most of the continuous wavelets are used for both wavelet decomposition and composition transforms. That is they are the continuous counterpart of
orthogonal wavelet
An orthogonal wavelet is a wavelet whose associated wavelet transform is orthogonal.
That is, the inverse wavelet transform is the adjoint of the wavelet transform.
If this condition is weakened one may end up with biorthogonal wavelets.
Basics ...
s.
The following continuous wavelets have been invented for various applications:
*
Poisson wavelet In mathematics, in functional analysis, several different wavelets are known by the name Poisson wavelet. In one context, the term "Poisson wavelet" is used to denote a family of wavelets labeled by the set of positive integers, the members of wh ...
*
Morlet wavelet
In mathematics, the Morlet wavelet (or Gabor wavelet)0).
The parameter \sigma in the Morlet wavelet allows trade between time and frequency resolutions. Conventionally, the restriction \sigma>5 is used to avoid problems with the Morlet wavelet a ...
Mexican hat wavelet
In mathematics and numerical analysis, the Ricker wavelet
:\psi(t) = \frac \left(1 - \left(\frac\right)^2 \right) e^
is the negative normalizing constant, normalized second derivative of a Gaussian function, i.e., up to scale and normalization, t ...
Meyer wavelet
The Meyer wavelet is an orthogonal wavelet proposed by Yves Meyer. As a type of a continuous wavelet, it has been applied in a number of cases, such as in adaptive filters, fractal random fields, and multi-fault classification.
The Meyer wavelet i ...
*
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 ...
*
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_ ...
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 ...