Hermitian wavelets are a family of discrete and continuous wavelets used in the constant and discrete Hermite wavelet transforms. The n^\textrm{th} Hermitian wavelet is defined as the normalized n^\textrm{th} derivative of a Gaussian distribution for each positive n:[1]

\Psi_{n}(x)=(2n)^{-\frac{n}{2}}c_{n}\operatorname{He}_{n}\left(x\right)e^{-\frac{1}{2}x^{2}},

where \operatorname{He}_{n}(x) denotes the n^\textrm{th} probabilist's Hermite polynomial. Each normalization coefficient c_{n} is given by

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 \Psi\in L_{\rho, \mu}(-\infty, \infty) is said to be an admissible Hermite wavelet if it satisfies the admissibility condition:[2]

C_\Psi = \sum_{n=0}^{\infty}{\frac{\|\hat\Psi (n)\|^2}{\|n\|}} < \infty

where \hat \Psi (n) are the terms of the Hermite transform of \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.[3]

Examples

The first three derivatives of the Gaussian function with \mu=0,\;\sigma=1:

f(t) = \pi^{-1/4}e^{(-t^2/2)},

are:

\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 L^2 norms \lVert f' \rVert=\sqrt{2}/2, \lVert f'' \rVert=\sqrt{3}/2, \lVert f^{(3)} \rVert= \sqrt{30}/4.

Normalizing the derivatives yields three Hermitian wavelets:

\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}

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.