Hermitian wavelet

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

\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 nth Hermite polynomial.

The normalisation coefficient cn 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 CΨ 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.