Hermitian wavelet

Hermitian wavelets are a family of discrete and continuous wavelets used in the constant and discrete Hermite wavelet transforms. The $n^{\textrm {th}}$ Hermitian wavelet is defined as the normalized $n^{\textrm {th}}$ 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 $\operatorname {He} _{n}(x)$ denotes the $n^{\textrm {th}}$ probabilist's Hermite polynomial. Each 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 condition:^[2]

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

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

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]

Examples

The first three derivatives of the Gaussian function with $\mu =0,\;\sigma =1$ : $f(t)=\pi ^{-1/4}e^{(-t^{2}/2)},$ are: ${\begin{aligned}f'(t)&=-\pi ^{-1/4}te^{(-t^{2}/2)},\\f''(t)&=\pi ^{-1/4}(t^{2}-1)e^{(-t^{2}/2)},\\f^{(3)}(t)&=\pi ^{-1/4}(3t-t^{3})e^{(-t^{2}/2)},\end{aligned}}$ and their $L^{2}$ norms $\lVert f'\rVert ={\sqrt {2}}/2,\lVert f''\rVert ={\sqrt {3}}/2,\lVert f^{(3)}\rVert ={\sqrt {30}}/4$ .

Normalizing the derivatives yields three Hermitian wavelets: ${\begin{aligned}\Psi _{1}(t)&={\sqrt {2}}\pi ^{-1/4}te^{(-t^{2}/2)},\\\Psi _{2}(t)&={\frac {2}{3}}{\sqrt {3}}\pi ^{-1/4}(1-t^{2})e^{(-t^{2}/2)},\\\Psi _{3}(t)&={\frac {2}{15}}{\sqrt {30}}\pi ^{-1/4}(t^{3}-3t)e^{(-t^{2}/2)}.\end{aligned}}$