Hermitian wavelet

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

$\Psi_(t)=(2n)^c_He_\left(t\right)e^$

where $He_\left(\right)$ denotes the $n^\textrm$ Hermite polynomial.

The normalisation coefficient $c_$ is given by:

$c_ = \left(n^\Gamma(n+\frac)\right)^ = \left(n^\sqrt2^(2n-1)!!\right)^\quad n\in\mathbb.$

The prefactor $C_$ in the resolution of the identity of the continuous wavelet transform for this wavelet is given by:

$C_=\frac$

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 Gaussian function with $\mu=0, \sigma=1$:

$f(t) = \pi^e^$

the first 3 derivatives read as,

:$\begin f'(t) & = -\pi^te^ \\ f''(t) & = \pi^(t^2 - 1)e^\\ f^(t) & = \pi^(3t - t^3)e^ \end$

and their $L^2$ norms $, , f', , =\sqrt/2, , , f'', , =\sqrt/2, , , f^, , = \sqrt/4$

So the wavelets which are the negative normalized derivatives are:

:$\begin \Psi_(t) &= \sqrt\pi^te^\\ \Psi_(t) &=\frac\sqrt\pi^(1-t^2)e^\\ \Psi_(t) &= \frac\sqrt\pi^(t^3 - 3t)e^ \end$

See also

* Wavelet

External links

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

{{DEFAULTSORT:Hermitian Wavelet
Continuous wavelets