Hermitian wavelet

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

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

where in this case we consider the (probabilist) Hermite polynomial ${\displaystyle \operatorname {He} _{n}(x)}$. The normalization coefficient ${\displaystyle c_{n}}$ is given by,

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

The function ${\displaystyle \Psi \in L_{\rho ,\mu }(-\infty ,\infty )}$ is said to be an admissible Hermite wavelet if it satisfies the admissibility relation:[2]

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

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

The perfector ${\displaystyle C_{\Psi }}$ in the resolution of the identity of the continuous wavelet transform for this wavelet is given by the formula,

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

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]