Hermitian wavelet

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Hermitian wavelets are a family of discrete and continuous wavelets used in the constant and discrete Hermite wavelet transforms. The <math>n^\textrm{th}</math> Hermitian wavelet is defined as the normalized <math>n^\textrm{th}</math> derivative of a Gaussian distribution for each positive <math>n</math>:<ref>Template:Cite journal</ref><math display="block">\Psi_{n}(x)=(2n)^{-\frac{n}{2}}c_{n}\operatorname{He}_{n}\left(x\right)e^{-\frac{1}{2}x^{2}}, </math>where <math>\operatorname{He}_{n}(x)</math> denotes the <math>n^\textrm{th}</math> probabilist's Hermite polynomial. Each normalization coefficient <math>c_{n}</math> is given by <math display="block">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}.</math> The function <math>\Psi\in L_{\rho, \mu}(-\infty, \infty)</math> is said to be an admissible Hermite wavelet if it satisfies the admissibility condition:<ref>Template:Cite journal</ref>

<math display="block">C_\Psi = \sum_{n=0}^{\infty}{\frac{\|\hat\Psi (n)\|^2}{\|n\|}} < \infty</math>

where <math>\hat \Psi (n)</math> are the terms of the Hermite transform of <math>\Psi</math>.

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.<ref>Template:Cite book</ref>

ExamplesEdit

The first three derivatives of the Gaussian function with <math>\mu=0,\;\sigma=1</math>:<math display="block">f(t) = \pi^{-1/4}e^{(-t^2/2)},</math>are:<math display="block">\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}</math>and their <math>L^2</math> norms <math>\lVert f' \rVert=\sqrt{2}/2, \lVert f \rVert=\sqrt{3}/2, \lVert f^{(3)} \rVert= \sqrt{30}/4</math>.

Normalizing the derivatives yields three Hermitian wavelets:<math display="block">\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}</math>

See alsoEdit

ReferencesEdit

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External linksEdit