Hermitian wavelet

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

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

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

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

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

See also

• Wavelet
• The Ricker wavelet is the ${\displaystyle n=2}$ Hermitian wavelet