Hermitian wavelet

Hermitian wavelets are a family of discrete and continuous wavelets, used in the continuous and discrete Hermite wavelet transform. The $n^{\textrm {th}}$ Hermitian wavelet is defined as the $n^{\textrm {th}}$ derivative of a Gaussian distribution for each positive $n$ :^[1]

\Psi _{n}(x)=(2n)^{-{\frac {n}{2}}}c_{n}\operatorname {He} _{n}\left(x\right)e^{-{\frac {1}{2}}x^{2}},

where in this case we consider the (probabilist) Hermite polynomial

\operatorname {He} _{n}(x)

The normalization coefficient $c_{n}$ is given by,

c_{n}=\left(n^{{\frac {1}{2}}-n}\Gamma \left(n+{\frac {1}{2}}\right)\right)^{-{\frac {1}{2}}}=\left(n^{{\frac {1}{2}}-n}{\sqrt {\pi }}2^{-n}(2n-1)!!\right)^{-{\frac {1}{2}}}\quad n\in \mathbb {N} .

The function

\Psi \in L_{\rho ,\mu }(-\infty ,\infty )

is said to be an admissible Hermite wavelet if it satisfies the admissibility relation:^[2]

C_{\Psi }=\sum _{n=0}^{\infty }{\frac {\|{\hat {\Psi }}(n)\|^{2}}{\|n\|}}<\infty

where ${\hat {\Psi }}(n)$ is the Hermite transform of $\Psi$ .

The perfector $C_{\Psi }$ in the resolution of the identity of the continuous wavelet transform for this wavelet is given by the formula,^{[further explanation needed]}

C_{\Psi }={\frac {4\pi n}{2n-1}}

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.^[3]