Hermitian Wavelet

Hermitian wavelets are a family of continuous wavelets, used in the continuous wavelet transform. The Hermitian wavelet is defined as the derivative of a Gaussian:

where denotes the Hermite polynomial.

The normalisation coefficient is given by:

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

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

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 :

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 norms

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}

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