Biomedical Engineering Reference
In-Depth Information
V
=
W
⊕
W
⊕
⊕
W
⊕
V
,
(2.9)
j
j
−
1
j
−
1
0
0
where
⊕
denotes the union of two orthogonal sets.
The DWT, using a dyadic scaling, is often used because of its practical effectiveness. The out-
put of the DWT forms a triangle to represent all resolution scales because of decimation (holding
one sample out of every two) and has the advantage of reducing the computational complexity and
storage. However, it is not possible to obtain representation with different scales at every instance
with the decimated output. This problem can be overcome by a nondecimated DWT [
61
], which
requires more computations and storage. The nondecimated DWT can be formed in two ways: (1)
The successive resolutions are obtained by the convolution between a given signal and an incre-
mental dilated wavelet function. (2) The successive resolutions are formed by smoothing with an
incremental dilated scaling function and taking the difference between successive smoothed data.
The à trous wavelet transform follows the latter procedure to produce a multiresolution
representation of the data. In this transform, successive convolutions with a discrete filter
h
is per-
formed as
¥
å
j
(2.10)
v
( )
k
=
h l v
( )
(
k
+
2
l
),
j
+
1
j
l
=-¥
where
v
0
(
k
) =
x
(
k
) is the original discrete time series, and the difference between successive smoothed
outputs is computed as
w
( )
k
=
v
( )
k
-
v
( ),
k
j
j
-1
j
(2.11)
where
w
j
represents the wavelet coefficients. It is clear that the original time series
x
(
k
) can be de-
composed as
S
å
1
(2.12)
x k
( )
=
v
( )
k
+
w
( ),
k
S
j
j
=
where
S
is the number of scales. The computational complexity of this algorithm is
O
(
N
) for data
length
N
.
Note that the à trous wavelet transform as originally defined by Sheppard does not necessarily
account for a causal decomposition (i.e., where the future data are not incorporated in the present
computation of the wavelet transform). To apply the à trous wavelet transform with such a require-
ment, the Haar à trous wavelet transform (HatWT) was introduced [
62
]. The impulse response
h
is
