Hermitian wavelets are a family of discrete and continuous wavelets used in the constant and discrete Hermite wavelet transforms. The n th {\displaystyle n^{\textrm {th}}} Hermitian wavelet is defined as the normalized n th {\displaystyle n^{\textrm {th}}} derivative of a Gaussian distribution for each positive n {\displaystyle n} : Ψ n ( x ) = ( 2 n ) − n 2 c n He n ( x ) e − 1 2 x 2 , {\displaystyle \Psi _{n}(x)=(2n)^{-{\frac {n}{2}}}c_{n}\operatorname {He} _{n}\left(x\right)e^{-{\frac {1}{2}}x^{2}},} where He n ( x ) {\displaystyle \operatorname {He} _{n}(x)} denotes the n th {\displaystyle n^{\textrm {th}}} probabilist's Hermite polynomial. Each normalization coefficient c n {\displaystyle c_{n}} is given by c n = ( n 1 2 − n Γ ( n + 1 2 ) ) − 1 2 = ( n 1 2 − n π 2 − n ( 2 n − 1 ) ! ! ) − 1 2 n ∈ N . {\displaystyle c_{n}=\left(n^{{\frac {1}{2}}-n}\Gamma \left(n+{\frac {1}{2}}\right)\right)^{-{\frac {1}{2}}}=\left(n^{{\frac {1}{2}}-n}{\sqrt {\pi }}2^{-n}(2n-1)!!\right)^{-{\frac {1}{2}}}\quad n\in \mathbb {N} .} The function Ψ ∈ L ρ , μ ( − ∞ , ∞ ) {\displaystyle \Psi \in L_{\rho ,\mu }(-\infty ,\infty )} is said to be an admissible Hermite wavelet if it satisfies the admissibility condition:
C Ψ = ∑ n = 0 ∞ ‖ Ψ ^ ( n ) ‖ 2 ‖ n ‖ < ∞ {\displaystyle C_{\Psi }=\sum _{n=0}^{\infty }{\frac {\|{\hat {\Psi }}(n)\|^{2}}{\|n\|}}<\infty }
where Ψ ^ ( n ) {\displaystyle {\hat {\Psi }}(n)} are the terms of the Hermite transform of Ψ {\displaystyle \Psi } . 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.
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