Hermitian wavelet

Hermitian wavelets are a family of discrete and continuous wavelets used in the constant and discrete Hermite wavelet transforms. The $n^{\text{th}}$ Hermitian wavelet is defined as the normalized $n^{\text{th}}$ derivative of a Gaussian distribution for each positive $n$:$${\mathrm{\Psi }}_n(x)=(2n{)}^{-\frac{n}{2}}{c}_n{\mathrm{He}}_n\left(x\right)e^{-\frac{1}{2}{x}^2},$$where ${\mathrm{He}}_n(x)$ denotes the $n^{\text{th}}$ probabilist's Hermite polynomial. Each normalization coefficient $c_n$ is given by $$c_n={\left(n^{\frac{1}{2}-n}\mathrm{\Gamma }(n+\frac{1}{2})\right)}^{-\frac{1}{2}}={(n^{\frac{1}{2}-n}\sqrt{\pi }{2}^{-n}(2n-1)!!)}^{-\frac{1}{2}}n\in \mathbb{N}.$$ The function $\mathrm{\Psi }\in L_{\rho ,\mu }(-\mathrm{\infty },\mathrm{\infty })$ is said to be an admissible Hermite wavelet if it satisfies the admissibility condition:

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

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

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.

Examples

The first three derivatives of the Gaussian function with $\mu =0,\sigma =1$:$$f(t)={\pi }^{-1/4}{e}^{(-t^2/2)},$$are:$$\begin{array}{cc}{f}^{\prime }(t)& =-{\pi }^{-1/4}t{e}^{(-t^2/2)},\\ f^{″}(t)& ={\pi }^{-1/4}(t^2-1)e^{(-t^2/2)},\\ f^{(3)}(t)& ={\pi }^{-1/4}(3t-t^3)e^{(-t^2/2)},\end{array}$$and their $L^2$ norms $\Vert f^{\prime }\Vert =\sqrt{2}/2,\Vert f^{″}\Vert =\sqrt{3}/2,\Vert f^{(3)}\Vert =\sqrt{30}/4$.

Normalizing the derivatives yields three Hermitian wavelets:$$\begin{array}{cc}{\mathrm{\Psi }}_1(t)& =\sqrt{2}{\pi }^{-1/4}t{e}^{(-t^2/2)},\\ {\mathrm{\Psi }}_2(t)& =\frac{2}{3}\sqrt{3}{\pi }^{-1/4}(1-t^2)e^{(-t^2/2)},\\ {\mathrm{\Psi }}_3(t)& =\frac{2}{15}\sqrt{30}{\pi }^{-1/4}(t^3-3t)e^{(-t^2/2)}.\end{array}$$

See also

• Wavelet
• The Ricker wavelet is the $n=2$ Hermitian wavelet

External links

• Hermitian Clifford–Hermite Wavelets (Department of Mathematical Analysis, Faculty of Engineering, Ghent University)