Hermitian wavelets are a family of discrete and continuous wavelets used in the constant and discrete Hermite wavelet transforms.
The
th
Hermitian wavelet is defined as the normalized
n
th
derivative of a Gaussian distribution for each positive
denotes the
probabilist's Hermite polynomial.
Each normalization coefficient
π
The function
ρ , μ
is said to be an admissible Hermite wavelet if it satisfies the admissibility condition:[2]
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] The first three derivatives of the Gaussian function with
μ = 0 ,
σ = 1
π
π
π
π
{\displaystyle {\begin{aligned}f'(t)&=-\pi ^{-1/4}te^{(-t^{2}/2)},\\f''(t)&=\pi ^{-1/4}(t^{2}-1)e^{(-t^{2}/2)},\\f^{(3)}(t)&=\pi ^{-1/4}(3t-t^{3})e^{(-t^{2}/2)},\end{aligned}}}
norms
Normalizing the derivatives yields three Hermitian wavelets:
π
π
π
{\displaystyle {\begin{aligned}\Psi _{1}(t)&={\sqrt {2}}\pi ^{-1/4}te^{(-t^{2}/2)},\\\Psi _{2}(t)&={\frac {2}{3}}{\sqrt {3}}\pi ^{-1/4}(1-t^{2})e^{(-t^{2}/2)},\\\Psi _{3}(t)&={\frac {2}{15}}{\sqrt {30}}\pi ^{-1/4}(t^{3}-3t)e^{(-t^{2}/2)}.\end{aligned}}}