Hermitian wavelet

Hermitian wavelets are a family of continuous wavelets, used in the continuous wavelet transform. The $n^\; extrm\{th\}$ Hermitian wavelet is defined as the $n^\; extrm\{th\}$ derivative of a Gaussian:

$Psi\_\{n\}(t)=(2n)^\{-frac\{n\}\{2c\_\{n\}H\_\{n\}left(frac\{t\}\{sqrt\{n\; ight)e^\{-frac\{1\}\{2n\}t^\{2$

where $H\_\{n\}left(\{x\}\; ight)$ denotes the $n^\; extrm\{th\}$ Hermite polynomial.

The normalisation coefficient $c\_\{n\}$ is given by:

$c\_\{n\}\; =\; left(n^\{frac\{1\}\{2\}-n\}Gamma(n+frac\{1\}\{2\})\; ight)^\{-frac\{1\}\{2\; =\; left(n^\{frac\{1\}\{2\}-n\}sqrt\{pi\}2^\{-n\}(2n-1)!!\; ight)^\{-frac\{1\}\{2quad\; ninmathbb\{Z\}.$

The prefactor $C\_\{Psi\}$ in the resolution of the identity of the continuous wavelet transform for this wavelet is given by:

$C\_\{Psi\}=frac\{4pi\; n\}\{2n-1\}$

i.e. Hermitian wavelets are admissible for all positive $n$.

In computer vision and image processing, Gaussian derivative operators of different orders are frequently used as a basis for expressing various types of visual operations; see scale-space and N-jet.