Graphics Reference
In-Depth Information
1
√
2
πσ
e
−
t
2
2
σ
2
,
g
σ
(
t
)=
1
2
√
πα
t
2
4
α
,
e
−
σ
2
=2
α.
=
letting
The Gaussian wavelet is, therefore,
t
4
α
√
πα
t
2
4
α
.
e
−
ψ
(
t
)=
−
Its Fourier transform is
ψ
(
ω
)=
∞
−∞
ψ
α
(
t
)
e
−iωt
dt,
=
iω e
−αω
2
.
Morlet Wavelet
The Morlet wavelet uses a windowed complex exponential. This was proposed
in [69] for signal analysis and is given by
1
√
2
π
e
−iω
o
t
e
−t
2
/
2
.
ψ
(
t
)=
Its Fourier transform is
ψ
(
ω
)=
e
−
(
ω−ω
o
)
2
/
2
,
where
ω
o
is the center frequency and the factor 1
/
√
2
π
guarantees
=1.
The center frequency
ω
o
is normally so chosen that the second maximum of
the real p
art
of
ψ
(
t
)
,t >
0 is half of the first one at
t
= 0. This provides
ψ
(
t
)
ω
0
=
π
2
ln
2
=5
.
336. One can notice that Morlet wavelet is not admissible
since
ψ
(0)
= 0. But it does not present any problem in practice since its value
is very small, roughly,
ψ
(0)
7
.
10
−
7
.
An important topic in wavelet theory is the discretization of the continuous
wavelet transform,
T
cw
(
f
(
a, b
)). We would like to have the wavelet
ψ
such that
f
can be recovered from
T
cw
(
f
(
a, b
)) values on a certain grid in the (
a, b
) plane,
i.e., from the values
≈
T
cw
(
f
(2
−j
,
2
−j
k
))
,
Z.
Note that
ψ
should have a property that the wavelets
2
j/
2
ψ
(2
j
j, k
∈
Z
constitute an orthonormal basis of
L
2
(
IR
). The Mexican hat or Marr wavelet
does not have this property. Such a function
ψ
is called the mother wavelet.
Often prior to the construction of the mother wavelet
ψ
, one constructs a func-
tion
φ
such that the functions
x
−
k
)
,
j, k
∈
Z
constitute an orthonormal
system.
φ
is, sometimes, called the father wavelet. This orthonormal system
then can be supplemented to a full orthonormal basis of
L
2
(
IR
) with the
functions
{
φ
(
t
−
k
)
}
,k
∈
2
j/
2
ψ
(2
j
t
−
k
)
,
j
∈
Z
+
,k
∈
Z.
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