Hermitian wavelet

Family of continuous wavelets
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Hermitian wavelets are a family of discrete and continuous wavelets, used in the continuous and discrete hermite wavelet transform. The n th {\displaystyle n^{\textrm {th}}} Hermitian wavelet is defined as the n th {\displaystyle n^{\textrm {th}}} derivative of a Gaussian distribution, for each positive n {\displaystyle n} :[1]

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 in this case we consider the "probabilist's Hermite polynomial" , He n ⁡ ( x ) {\displaystyle \operatorname {He} _{n}(x)} .

The 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 relation:[2]

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)} is the Hermite transform of Ψ {\displaystyle \Psi } .

The perfector C Ψ {\displaystyle C_{\Psi }} in the resolution of the identity of the continuous wavelet transform for this wavelet is given by the formula,[further explanation needed]

C Ψ = 4 π n 2 n − 1 {\displaystyle C_{\Psi }={\frac {4\pi n}{2n-1}}}
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.[3]

Examples

The first three derivatives of the Gaussian function with μ = 0 , σ = 1 {\displaystyle \mu =0,\;\sigma =1} :

f ( t ) = π − 1 / 4 e ( − t 2 / 2 ) , {\displaystyle f(t)=\pi ^{-1/4}e^{(-t^{2}/2)},}
are:
f ′ ( t ) = − π − 1 / 4 t e ( − t 2 / 2 ) , f ″ ( t ) = π − 1 / 4 ( t 2 − 1 ) e ( − t 2 / 2 ) , f ( 3 ) ( t ) = π − 1 / 4 ( 3 t − t 3 ) e ( − t 2 / 2 ) , {\displaystyle {\begin{aligned}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{aligned}}}
and their L 2 {\displaystyle L^{2}} norms | | f ′ | | = 2 / 2 , | | f ″ | | = 3 / 2 , | | f ( 3 ) | | = 30 / 4 {\displaystyle ||f'||={\sqrt {2}}/2,||f''||={\sqrt {3}}/2,||f^{(3)}||={\sqrt {30}}/4} .

Normalizing the derivatives yields three Hermitian wavelets:

Ψ 1 ( t ) = 2 π − 1 / 4 t e ( − t 2 / 2 ) , Ψ 2 ( t ) = 2 3 3 π − 1 / 4 ( 1 − t 2 ) e ( − t 2 / 2 ) , Ψ 3 ( t ) = 2 15 30 π − 1 / 4 ( t 3 − 3 t ) e ( − t 2 / 2 ) . {\displaystyle {\begin{aligned}\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{aligned}}}

See also

References

  1. ^ Brackx, F.; De Schepper, H.; De Schepper, N.; Sommen, F. (2008-02-01). "Hermitian Clifford-Hermite wavelets: an alternative approach". Bulletin of the Belgian Mathematical Society, Simon Stevin. 15 (1). doi:10.36045/bbms/1203692449. ISSN 1370-1444.
  2. ^ "Continuous and Discrete Wavelet Transforms Associated with Hermite Transform". International Journal of Analysis and Applications. 2020. doi:10.28924/2291-8639-18-2020-531.
  3. ^ Wah, Benjamin W., ed. (2007-03-15). Wiley Encyclopedia of Computer Science and Engineering (1 ed.). Wiley. doi:10.1002/9780470050118.ecse609. ISBN 978-0-471-38393-2.

External links