Graphics Reference
In-Depth Information
•
Scale: If
f
(
t
) has a continuous wavelet transform
T
cw
(
f
(
a, b
)), then the con-
tinuous wavelet transform of the scaled function
√
s
f
(
s
,
s
)is
T
cw
(
f
(
s
,
s
)).
1
√
s
f
(
s
,
s
)
1
The continuous wavelet transform of
+
∞
s
)
dt
=
s
+
∞
ψ
(
t
−
b
)
f
(
t
ψ
(
sT
−
b
√
|a|
1
)
f
(
T
)
dT,
a
|
a
|
a
s
−∞
−∞
=
s
|
+
∞
b
s
ψ
(
T
−
)
f
(
T
)
dT,
a
s
a
|
−∞
=
T
cw
(
f
(
s
,
s
)
,
t
s
where we let
=
T
. Thus, when the function is scaled, its
T
cw
is also
scaled.
•
Energy of conservation: Continuous wavelet transform has an energy con-
servation property similar to that of Fourier transform.
•
Localization: The continuous wavelet transform has sharp time localization
at high frequencies and this distinguishes the wavelet transform from the
traditional Fourier or Fourier-like transform.
•
Time localization: To check the time localization of a particular wavelet,
one can examine the wavelet transform of a Dirac pulse using the wavelet
in question. For a given scale factor, the transform is equal to the scaled
wavelet reversed in time and centered at the location of the Dirac.
8.4 A Glimpse of Continuous Wavelets
Continuous wavelets can be viewed in two different forms, isotropic and
anisotropic wavelets, depending on how they can be applied in real life prob-
lems. For point-wise analysis, i.e., when no oriented features are present or
relevant in the signal, we may choose an analyzing wavelet
ψ
, which is invari-
ant under rotation. A typical example of an isotropic wavelet is the Mexican
hat wavelet. But when directional features are in the signal or when one is
interested in directional filtering, anisotropic wavelets are of much use. Typ-
ical directional or anisotropic wavelets are Morelet wavelet or the Cauchy
wavelets. Whether isotropic or anisotropic, these are all the basic wavelets.
8.4.1 Basic Wavelets
Below, we describe two important basic wavelets.
Gaussian Wavelet
A Gaussian wavelet is simply the derivative of Gaussian function. The Gaus-
sian function is
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