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)
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