Hermitian wavelet

Hermitian wavelets are a family of continuous wavelets, used in the continuous wavelet transform. The n th Hermitian wavelet is defined as the n th derivative of a Gaussian distribution:

Ψ n ( t ) = ( 2 n ) − n 2 c n H n ( t n ) e − 1 2 n t 2

where H n ( x ) denotes the n th Hermite polynomial.

The normalisation coefficient c n is given by:

c n = ( n 1 2 − n Γ ( n + 1 2 ) ) − 1 2 = ( n 1 2 − n π 2 − n ( 2 n − 1 ) ! ! ) − 1 2 n ∈ Z .

The prefactor C Ψ in the resolution of the identity of the continuous wavelet transform for this wavelet is given by:

C Ψ = 4 π n 2 n − 1

i.e. Hermitian wavelets are admissible for all positive n .

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.

Examples of Hermitian wavelets: Starting from a Gaussian function with μ = 0 , σ = 1 :

f ( t ) = π − 1 / 4 e ( − t 2 / 2 )

the first 3 derivatives read

and their L 2 norms | | f ′ | | = 2 / 2 , | | f ″ | | = 3 / 2 , | | f ( 3 ) | | = 30 / 4

So the wavelets which are the negative normalized derivatives are: