Family of continuous wavelets
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's 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]
References[edit]
External links[edit]