Environmental Engineering Reference
In-Depth Information
signal
f
. Note that in the CWT , the values of
b
are selected to cover the entire
signal (
i.e.
, space or length of time covered by
f
(
x
)
), but
a
can take any values.
To reduce computing time and the amount of data generated by this analysis,
the discrete wavelet transform (DWT) was proposed, which is performed only at
a smaller number of discrete locations and frequencies. It was shown that select-
ing shifting and dilations (
i.e.
, discrete frequencies) as a power of 2, called dyadic,
would yield great computational economy with similar accuracy as the CWT. The
DWT using dyadic locations and frequencies is obtained by setting 1
(
x
)
2
−
m
/
a
=
or
2
m
and
n
2
m
, where both
m
and
n
are integers.
a
=
−
b
/
a
= −
n
or
b
=
2
−
m
/
2
ψ
2
−
m
x
ψ
m
(
x
)=
(
−
n
).
(3.9)
,
n
The wavelet coefficients are computed as for the CWT but only at specific
m
and
n
values:
=
(
)
(
)
=
(
),
(
).
d
m
,
n
f
x
ψ
m
,
n
x
dx
ψ
m
,
n
x
f
x
(3.10)
ℜ
Practical implementation of DWT algorithms are, however, performed using
quadrature mirror filters. Indeed, a very important contribution of Mallat was to link
orthogonal wavelets to the filters traditionally used in the signal processing field
[47]. Through this work, a new function called
scaling function
φ
(
x
)
, orthogonal to
the wavelet function ψ
, was introduced to capture the low frequency information
(
i.e.
coarse) whereas the high frequency detail is extracted by the wavelet function.
These two functions are expressed in terms of orthogonal filters, a low-pass filter
h
0
related to the scaling function φ
(
x
)
(
x
)
, and a high-pass filter
h
1
, related to the wavelet
function ψ
(
x
)
, as shown below:
2
j
/
2
h
1
2
j
l
2
j
/
2
h
0
2
j
l
ψ
j
[
k
]=
[
k
−
]
and φ
j
[
k
]=
[
k
−
],
(3.11)
,
l
,
l
where
j
is the decomposition level,
l
is the shifting parameter, and
k
is the discrete
location, either in time or space, on the signal to be decomposed (all integers).
From these two functions, the wavelet coefficients can be computed as the in-
ner product (or scalar product) of the signal
f
[
]
and the wavelet and the scaling
functions. This results in two sets of coefficients, those capturing the low frequency
information (
i.e.
, associated with the low-pass filter), called the approximation co-
efficients
a
k
, and the second set representing the details or high-frequency in-
formation called the detail coefficients
d
)
[
k
]
(
j
)
[
k
]
:
(
j
a
)
[
k
]=
f
[
k
],
φ
j
,
l
[
k
]
and
d
)
[
k
]=
f
[
k
],
ψ
j
,
l
[
k
].
(3.12)
(
j
(
j
These coefficients are calculated sequentially, for each level of decomposition
j
,us-
ing filter banks [46]. How to implement wavelet series expansion using filter banks
was shown by Mallat [41], while Daubechies [48, 49] showed how to construct
wavelets from filter banks [46].
Applying DWT to a multidimensional signal, such as a 2-D image, can be readily
accomplished by using 1-D DWT along each dimension, one at a time [46]. This is
referred to as a
separable solution
, which uses separable filters (
i.e.
,
h
0
and
h
1
within
Search WWH ::
Custom Search