Hermitian wavelet explained

Hermitian wavelets are a family of discrete and continuous wavelets used in the constant and discrete Hermite wavelet transforms. The

n^rm{th}

Hermitian wavelet is defined as the normalized

n^rm{th}

derivative of a Gaussian distribution for each positive

:^[1]

\Psi_(x)=(2n)^c_\operatorname_\left(x\right)e^,

where

\operatorname{He}_n(x)

denotes the

n^rm{th}

probabilist's Hermite polynomial. Each normalization coefficient

c_n

is given by

c_ = \left(n^\Gamma\left(n+\frac\right)\right)^ = \left(n^\sqrt2^(2n-1)!!\right)^\quad n\in\mathbb.

The function

\Psi\inL_\rho,(-infty,infty)

is said to be an admissible Hermite wavelet if it satisfies the admissibility condition:^[2]

$C_\Psi = \sum_^ < \infty$

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^e^,

are:

\begin f'(t) & = -\pi^te^, \\ f

(t) & = \pi^(t^2 - 1)e^,\\f^(t) & = \pi^(3t - t^3)e^, \endand their
L²

norms
\lVertf'\rVert=\sqrt{2}/2,\lVertf''\rVert=\sqrt{3}/2,\lVertf⁽³⁾\rVert=\sqrt{30}/4

.

Normalizing the derivatives yields three Hermitian wavelets: $\begin\Psi_(t) &= \sqrt\pi^te^,\\\Psi_(t) &=\frac\sqrt\pi^(1-t^2)e^,\\\Psi_(t) &= \frac\sqrt\pi^(t^3 - 3t)e^.\end$

External links

Hermitian Clifford–Hermite Wavelets (Department of Mathematical Analysis, Faculty of Engineering, Ghent University)

Notes and References

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. 10.36045/bbms/1203692449. 1370-1444. free.
2020 . Continuous and Discrete Wavelet Transforms Associated with Hermite Transform . International Journal of Analysis and Applications . 10.28924/2291-8639-18-2020-531. free .
Book: Wiley Encyclopedia of Computer Science and Engineering . 2007-03-15 . Wiley . 978-0-471-38393-2 . Wah . Benjamin W. . 1 . en . 10.1002/9780470050118.ecse609.

Hermitian wavelet explained

Examples

See also

External links

Notes and References