Biomedical Engineering Reference
In-Depth Information
It is easy to prove that the total amount of data after a discrete wavelet expan-
sion as shown in Fig. 6.2 has the same length to the input signal. Therefore, such
transform provides a compact representation of the signal suited to data com-
pression as wavelet transform provides a better spatial-frequency localization.
On the other hand, since the data was downsampled at each level of expan-
sion, such transform performs poorly on localization or detection problems.
Mathematically, the transform is variant under translation of the signal (i.e. is
dependent on the downsampling scheme used during the decomposition), which
makes it less attractive for analysis of nonstationary signals. In image analysis,
translation invariance is critical to the preservation of all the information of the
signal and a redundant representation needs to be applied.
In the dyadic wavelet transform framework proposed by Mallat
et al.
[16],
sampling of the translation parameter was performed with the same sampling
period as that of the input signal in order to preserve translation invariance.
A more general framework of wavelet transform can be designed with dif-
ferent reconstruction and decomposition filters that form a biorthogonal basis.
Such generalization provides more flexibility in the design of the wavelet func-
tions. In that case, similar to Eq. (6.11), the discrete dyadic wavelet transform
of a signal
s
(
n
) is defined as a sequence of discrete functions:
{
S
M
s
(
n
)
,
{
W
m
s
(
n
)
}
m
∈
[
I
,
M
]
}
n
∈Z
,
(6.14)
where
S
M
s
(
n
)
=
s
∗
φ
M
(
n
) represents the DC component, or the coarsest infor-
mation from the input signal.
Given a pair of wavelet function
ψ
(
x
) and reconstruction function
χ
(
x
), the
discrete dyadic wavelet transform (decomposition and reconstruction) can be
implemented with a fast filter bank scheme using a pair of decomposition filters
H
,
G
and a reconstruction filter
K
[16]:
φ
(2
ω
)
=
e
−
i
ω
s
H
(
ω
)
ˆ
ˆ
φ
(
ω
)
,
ψ
(2
ω
)
=
e
−
i
ω
s
G
(
ω
)
ˆ
ˆ
ψ
(
ω
)
,
(6.15)
χ
(2
ω
)
=
e
i
ω
s
K
(
ω
)
χ
(
ω
)
,
where
s
is a
ψ
(
x
)-dependent sampling shift. The three filters satisfy:
2
|
H
(
ω
)
|
+
G
(
ω
)
K
(
ω
)
=
1
.
(6.16)
Defining
F
s
(
ω
)
=
e
−
i
ω
s
F
(
ω
), where
F
is
H
,
G
,or
K
, we can construct a filter
bank implementation of the discrete dyadic wavelet transform as illustrated in
