Hermitian wavelet

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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:

\Psi_{n}(t)=(2n)^{-\frac{n}{2}}c_{n}H_{n}\left(\frac{t}{\sqrt{n}}\right)e^{-\frac{1}{2n}t^{2}}

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

The normalisation coefficient c_{n} is given by:

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

C_{\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{align}
            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{align}

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

So the wavelets which are the negative normalized derivatives are:

\begin{align}
\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{align}