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 :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:
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.
The first three derivatives of the Gaussian function with :are:and their norms .
Normalizing the derivatives yields three Hermitian wavelets: