Hermitian wavelet

Hermitian wavelets are a family of continuous wavelets, used in the continuous wavelet transform. The $n^{\text{<mrow data-mjx-texclass="ORD">th</mrow>}}$ Hermitian wavelet is defined as the $n^{\text{<mrow data-mjx-texclass="ORD">th</mrow>}}$ derivative of a Gaussian distribution:

${\mathrm{\Psi }}_n(t)=(2n{)}^{-\frac{n}{2}}{c}_{n}H{e}_n\left(t\right)e^{-\frac{1}{2n}{t}^2}$

where $H{e}_n\left(x\right)$ denotes the $n^{\text{<mrow data-mjx-texclass="ORD">th</mrow>}}$ Hermite polynomial.

The normalisation coefficient $c_n$ is given by:

$c_n={(n^{\frac{1}{2}-n}\mathrm{\Gamma }(n+\frac{1}{2}))}^{-\frac{1}{2}}={(n^{\frac{1}{2}-n}\sqrt{\pi }{2}^{-n}(2n-1)!!)}^{-\frac{1}{2}}n\in \mathbb{\mathbb{Z}}.$

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

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

i.e. Hermitian wavelets are admissible for all positive $n$.

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 $\mu =0,\sigma =1$:

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

the first 3 derivatives read

$\begin{array}{rl}{f}^{\prime }(t)& =-{\pi }^{-1/4}t{e}^{(-t^2/2)}\\ f^{″}(t)& ={\pi }^{-1/4}(t^2-1)e^{(-t^2/2)}\\ f^{(3)}(t)& ={\pi }^{-1/4}(3t-t^3)e^{(-t^2/2)}\end{array}$

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

So the wavelets which are the negative normalized derivatives are:

$\begin{array}{rl}{\mathrm{\Psi }}_1(t)& =\sqrt{2}{\pi }^{-1/4}t{e}^{(-t^2/2)}\\ {\mathrm{\Psi }}_2(t)& =\frac{2}{3}\sqrt{3}{\pi }^{-1/4}(1-t^2)e^{(-t^2/2)}\\ {\mathrm{\Psi }}_3(t)& =\frac{2}{15}\sqrt{30}{\pi }^{-1/4}(t^3-3t)e^{(-t^2/2)}\end{array}$