- Source: Meyer wavelet
The Meyer wavelet is an orthogonal wavelet proposed by Yves Meyer. As a type of a continuous wavelet, it has been applied in a number of cases, such as in adaptive filters, fractal random fields, and multi-fault classification.
The Meyer wavelet is infinitely differentiable with infinite support and defined in frequency domain in terms of function
ν
{\displaystyle \nu }
as
Ψ
(
ω
)
:=
{
1
2
π
sin
(
π
2
ν
(
3
|
ω
|
2
π
−
1
)
)
e
j
ω
/
2
if
2
π
/
3
<
|
ω
|
<
4
π
/
3
,
1
2
π
cos
(
π
2
ν
(
3
|
ω
|
4
π
−
1
)
)
e
j
ω
/
2
if
4
π
/
3
<
|
ω
|
<
8
π
/
3
,
0
otherwise
,
{\displaystyle \Psi (\omega ):={\begin{cases}{\frac {1}{\sqrt {2\pi }}}\sin \left({\frac {\pi }{2}}\nu \left({\frac {3|\omega |}{2\pi }}-1\right)\right)e^{j\omega /2}&{\text{if }}2\pi /3<|\omega |<4\pi /3,\\{\frac {1}{\sqrt {2\pi }}}\cos \left({\frac {\pi }{2}}\nu \left({\frac {3|\omega |}{4\pi }}-1\right)\right)e^{j\omega /2}&{\text{if }}4\pi /3<|\omega |<8\pi /3,\\0&{\text{otherwise}},\end{cases}}}
where
ν
(
x
)
:=
{
0
if
x
<
0
,
x
if
0
<
x
<
1
,
1
if
x
>
1.
{\displaystyle \nu (x):={\begin{cases}0&{\text{if }}x<0,\\x&{\text{if }}0
There are many different ways for defining this auxiliary function, which yields variants of the Meyer wavelet.
For instance, another standard implementation adopts
ν
(
x
)
:=
{
x
4
(
35
−
84
x
+
70
x
2
−
20
x
3
)
if
0
<
x
<
1
,
0
otherwise
.
{\displaystyle \nu (x):={\begin{cases}x^{4}(35-84x+70x^{2}-20x^{3})&{\text{if }}0
The Meyer scale function is given by
Φ
(
ω
)
:=
{
1
2
π
if
|
ω
|
<
2
π
/
3
,
1
2
π
cos
(
π
2
ν
(
3
|
ω
|
2
π
−
1
)
)
if
2
π
/
3
<
|
ω
|
<
4
π
/
3
,
0
otherwise
.
{\displaystyle \Phi (\omega ):={\begin{cases}{\frac {1}{\sqrt {2\pi }}}&{\text{if }}|\omega |<2\pi /3,\\{\frac {1}{\sqrt {2\pi }}}\cos \left({\frac {\pi }{2}}\nu \left({\frac {3|\omega |}{2\pi }}-1\right)\right)&{\text{if }}2\pi /3<|\omega |<4\pi /3,\\0&{\text{otherwise}}.\end{cases}}}
In the time domain, the waveform of the Meyer mother-wavelet has the shape as shown in the following figure:
Close expressions
Valenzuela and de Oliveira give the explicit expressions of Meyer wavelet and scale functions:
ϕ
(
t
)
=
{
2
3
+
4
3
π
t
=
0
,
sin
(
2
π
3
t
)
+
4
3
t
cos
(
4
π
3
t
)
π
t
−
16
π
9
t
3
otherwise
,
{\displaystyle \phi (t)={\begin{cases}{\frac {2}{3}}+{\frac {4}{3\pi }}&t=0,\\{\frac {\sin({\frac {2\pi }{3}}t)+{\frac {4}{3}}t\cos({\frac {4\pi }{3}}t)}{\pi t-{\frac {16\pi }{9}}t^{3}}}&{\text{otherwise}},\end{cases}}}
and
ψ
(
t
)
=
ψ
1
(
t
)
+
ψ
2
(
t
)
,
{\displaystyle \psi (t)=\psi _{1}(t)+\psi _{2}(t),}
where
ψ
1
(
t
)
=
4
3
π
(
t
−
1
2
)
cos
[
2
π
3
(
t
−
1
2
)
]
−
1
π
sin
[
4
π
3
(
t
−
1
2
)
]
(
t
−
1
2
)
−
16
9
(
t
−
1
2
)
3
,
{\displaystyle \psi _{1}(t)={\frac {{\frac {4}{3\pi }}(t-{\frac {1}{2}})\cos[{\frac {2\pi }{3}}(t-{\frac {1}{2}})]-{\frac {1}{\pi }}\sin[{\frac {4\pi }{3}}(t-{\frac {1}{2}})]}{(t-{\frac {1}{2}})-{\frac {16}{9}}(t-{\frac {1}{2}})^{3}}},}
ψ
2
(
t
)
=
8
3
π
(
t
−
1
2
)
cos
[
8
π
3
(
t
−
1
2
)
]
+
1
π
sin
[
4
π
3
(
t
−
1
2
)
]
(
t
−
1
2
)
−
64
9
(
t
−
1
2
)
3
.
{\displaystyle \psi _{2}(t)={\frac {{\frac {8}{3\pi }}(t-{\frac {1}{2}})\cos[{\frac {8\pi }{3}}(t-{\frac {1}{2}})]+{\frac {1}{\pi }}\sin[{\frac {4\pi }{3}}(t-{\frac {1}{2}})]}{(t-{\frac {1}{2}})-{\frac {64}{9}}(t-{\frac {1}{2}})^{3}}}.}
References
Daubechies, Ingrid (September 1992). Ten Lectures on Wavelets (CBMS-NSF conference series in applied mathematics) (SIAM ed.). Springer-Verlag. pp. 117–119, 137–138, 152–155. ISBN 978-0-89871-274-2.
External links
wavelet toolbox
Matlab implementation
Kata Kunci Pencarian:
- Yves Meyer
- Penghargaan Abel
- Eksperimen semu
- Meyer wavelet
- Wavelet
- Continuous wavelet
- Yves Meyer
- Wavelet transform
- Stéphane Mallat
- Multiresolution analysis
- Conservatoire national des arts et métiers
- Shearlet
- Ingrid Daubechies
