Hermitian wavelet
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Hermitian wavelets are a family of continuous wavelets, used in the continuous wavelet transform. The Hermitian wavelet is defined as the derivative of a Gaussian distribution, for each positive :[1]
where denotes the probabilist's Hermite polynomial.
The normalization coefficient is given by,
The perfector in the resolution of the identity of the continuous wavelet transform for this wavelet is given by the formula,[further explanation needed]
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.[2]
Examples[edit]
The first three derivatives of the Gaussian function with :
are:
and their norms .
Normalizing the derivatives yields three Hermitian wavelets:
See also[edit]
- Wavelet
- The Ricker wavelet is the Hermitian wavelet
References[edit]
- ^ 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.
- ^ 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.