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Family of continuous wavelets
Hermitian wavelets are a family of discrete and
continuous wavelets , used in the continuous and discrete hermite wavelet transform. The
n
th
{\displaystyle n^{\textrm {th}}}
Hermitian wavelet is defined as the
n
th
{\displaystyle n^{\textrm {th}}}
derivative of a
Gaussian distribution , for each positive
n
{\displaystyle n}
:
[1]
Ψ
n
(
x
)
=
(
2
n
)
−
n
2
c
n
He
n
(
x
)
e
−
1
2
x
2
,
{\displaystyle \Psi _{n}(x)=(2n)^{-{\frac {n}{2}}}c_{n}\operatorname {He} _{n}\left(x\right)e^{-{\frac {1}{2}}x^{2}},}
where in this case we consider the
"probabilist's Hermite polynomial" ,
He
n
(
x
)
{\displaystyle \operatorname {He} _{n}(x)}
.
The normalization coefficient
c
n
{\displaystyle 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
∈
N
.
{\displaystyle c_{n}=\left(n^{{\frac {1}{2}}-n}\Gamma \left(n+{\frac {1}{2}}\right)\right)^{-{\frac {1}{2}}}=\left(n^{{\frac {1}{2}}-n}{\sqrt {\pi }}2^{-n}(2n-1)!!\right)^{-{\frac {1}{2}}}\quad n\in \mathbb {N} .}
The function
Ψ
∈
L
ρ
,
μ
(
−
∞
,
∞
)
{\displaystyle \Psi \in L_{\rho ,\mu }(-\infty ,\infty )}
is said to be an admissible Hermite wavelet if it satisfies the admissibility relation:
[2]
C
Ψ
=
∑
n
=
0
∞
‖
Ψ
^
(
n
)
‖
2
‖
n
‖
<
∞
{\displaystyle C_{\Psi }=\sum _{n=0}^{\infty }{\frac {\|{\hat {\Psi }}(n)\|^{2}}{\|n\|}}<\infty }
where
Ψ
^
(
n
)
{\displaystyle {\hat {\Psi }}(n)}
is the Hermite transform of
Ψ
{\displaystyle \Psi }
.
The perfector
C
Ψ
{\displaystyle C_{\Psi }}
in the resolution of the identity of the continuous wavelet transform for this wavelet is given by the
formula ,[
further explanation needed ]
C
Ψ
=
4
π
n
2
n
−
1
{\displaystyle C_{\Psi }={\frac {4\pi n}{2n-1}}}
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
The first three derivatives of the
Gaussian function with
μ
=
0
,
σ
=
1
{\displaystyle \mu =0,\;\sigma =1}
:
f
(
t
)
=
π
−
1
/
4
e
(
−
t
2
/
2
)
,
{\displaystyle f(t)=\pi ^{-1/4}e^{(-t^{2}/2)},}
are:
f
′
(
t
)
=
−
π
−
1
/
4
t
e
(
−
t
2
/
2
)
,
f
″
(
t
)
=
π
−
1
/
4
(
t
2
−
1
)
e
(
−
t
2
/
2
)
,
f
(
3
)
(
t
)
=
π
−
1
/
4
(
3
t
−
t
3
)
e
(
−
t
2
/
2
)
,
{\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}}}
and their
L
2
{\displaystyle L^{2}}
norms
|
|
f
′
|
|
=
2
/
2
,
|
|
f
″
|
|
=
3
/
2
,
|
|
f
(
3
)
|
|
=
30
/
4
{\displaystyle ||f'||={\sqrt {2}}/2,||f''||={\sqrt {3}}/2,||f^{(3)}||={\sqrt {30}}/4}
.
Normalizing the derivatives yields three Hermitian wavelets:
Ψ
1
(
t
)
=
2
π
−
1
/
4
t
e
(
−
t
2
/
2
)
,
Ψ
2
(
t
)
=
2
3
3
π
−
1
/
4
(
1
−
t
2
)
e
(
−
t
2
/
2
)
,
Ψ
3
(
t
)
=
2
15
30
π
−
1
/
4
(
t
3
−
3
t
)
e
(
−
t
2
/
2
)
.
{\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}}}
See also
References
External links