Hermitian wavelet

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

${\displaystyle \Psi _{n}(t)=(2n)^{-{\frac {n}{2}}}c_{n}He_{n}\left(t\right)e^{-{\frac {1}{2n}}t^{2}}}$

where ${\displaystyle He_{n}\left({x}\right)}$ denotes the ${\displaystyle n^{\textrm {th}}}$ Hermite polynomial.

The normalisation coefficient ${\displaystyle c_{n}}$ is given by:

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

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

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

i.e. Hermitian wavelets are admissible for all positive ${\displaystyle n}$.

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.

Examples of Hermitian wavelets: Starting from a Gaussian function with ${\displaystyle \mu =0,\sigma =1}$:

${\displaystyle f(t)=\pi ^{-1/4}e^{(-t^{2}/2)}}$

the first 3 derivatives read

${\displaystyle {\begin{aligned}f'(t)&=-\pi ^{-1/4}te^{(-t^{2}/2)}\\f''(t)&=\pi ^{-1/4}(t^{2}-1)e^{(-t^{2}/2)}\\f^{(3)}(t)&=\pi ^{-1/4}(3t-t^{3})e^{(-t^{2}/2)}\end{aligned}}}$

and their ${\displaystyle L^{2}}$ norms ${\displaystyle ||f'||={\sqrt {2}}/2,||f''||={\sqrt {3}}/2,||f^{(3)}||={\sqrt {30}}/4}$

So the wavelets which are the negative normalized derivatives are:

${\displaystyle {\begin{aligned}\Psi _{1}(t)&={\sqrt {2}}\pi ^{-1/4}te^{(-t^{2}/2)}\\\Psi _{2}(t)&={\frac {2}{3}}{\sqrt {3}}\pi ^{-1/4}(1-t^{2})e^{(-t^{2}/2)}\\\Psi _{3}(t)&={\frac {2}{15}}{\sqrt {30}}\pi ^{-1/4}(t^{3}-3t)e^{(-t^{2}/2)}\end{aligned}}}$