Hermitian wavelet

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

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

where $\operatorname {He} _{n}(x)$ denotes the $n^{\textrm {th}}$ probabilist's Hermite polynomial. 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 perfector $C_{\Psi }$ in the resolution of the identity of the continuous wavelet transform for this wavelet is given by the formula,

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.

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