Hermitian wavelet
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Hermitian wavelets are a family of discrete and continuous wavelets, used in the continuous and discrete Hermite wavelet transform. The Hermitian wavelet is defined as the derivative of a Gaussian distribution for each positive :[1]
where in this case we consider the (probabilist) Hermite polynomial .
The normalization coefficient is given by,
The function is said to be an admissible Hermite wavelet if it satisfies the admissibility relation:[2]
where is the Hermite transform of .
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.[3]
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.
- ^ "Continuous and Discrete Wavelet Transforms Associated with Hermite Transform". International Journal of Analysis and Applications. 2020. doi:10.28924/2291-8639-18-2020-531.
- ^ 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.