Environmental Engineering Reference
In-Depth Information
(a)
(b)
Figure 3.8
Brick wall image. (a) coarse pattern defined by the bricks and mortar joints. (b) fine
surface texture of a brick. From Tessier
et al.
[26]
all detail signals from levels 1 to
J
. Wavelet decomposition is one method satisfying
the MRA conditions and is frequently used for signal compression, denoising, but
also for texture analysis of 2-D signals (
i.e.
images).
In Fourier analysis, a continuous 1-D signal or function
f
(
e.g.
, a time series)
is decomposed into a series of periodic functions or waves of infinite duration rep-
resented by sines and cosines at different frequencies. The wavelet transform, on
the other hand, uses waveforms of
finite length
. This allows space/time-frequency
decomposition of
f
(
x
)
instead of frequency decomposition only as obtained by
the Fourier transform. Many waveforms are available for this purpose (
e.g.
Haar,
Daubechies, Coiflet, Symlet, Mexican hat,
etc.
), and each is mathematically repre-
sented by a function called the
mother wavelet
ψ
(
x
)
(
x
)
. Translation and dilation of the
mother wavelet ψ
,where
a
and
b
are in-
tegers, respectively scaling and translating the mother wavelet as described below.
Since the scaling coefficient
a
compresses or stretches wavelet ψ
a
(
x
)
generates a family of wavelets ψ
a
(
x
)
,
b
(
x
)
, it effectively
,
b
changes the frequency of the wavelet function.
ψ
x
1
|
−
b
ψ
a
,
b
(
x
)=
.
(3.7)
a
a
|
The continuous wavelet transform (CWT) is the convolution of signal
f
(
x
)
with
the scaled and dilated wavelet function ψ
a
,
b
resulting in a scalar quantity called
the wavelet coefficient
d
a
,
b
. The convolution integral can also be expressed as a
scalar product between the two functions represented by vectors:
(
x
)
ℜ
f
d
a
=
(
x
)
ψ
a
(
x
)
dx
=
ψ
a
(
x
),
f
(
x
).
(3.8)
,
b
,
b
,
b
Hence, the wavelet coefficient
d
a
b
can be viewed as a measure of similarity be-
,
tween the signal
f
[13], at a particular location
in space or time, determined by the value of
b
, and at a specific frequency, given
by the value of
a
. Applying this transform to several values of
a
and
b
yields a set
of wavelet coefficients characterizing the space/time-frequency decomposition of
(
x
)
and the wavelet function ψ
a
(
x
)
,
b
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