Hermitian wavelets are a family of discrete and continuous wavelets used in the constant and discrete Hermite wavelet transforms. The Hermitian wavelet is defined as the normalized derivative of a Gaussian distribution for each positive :[1]where denotes the probabilist's Hermite polynomial. Each normalization coefficient is given by The function is said to be an admissible Hermite wavelet if it satisfies the admissibility condition:[2]

where are the terms of the Hermite transform of .

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

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The first three derivatives of the Gaussian function with  : are: and their   norms  .

Normalizing the derivatives yields three Hermitian wavelets: 

See also

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References

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  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.
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