Biomedical Engineering Reference
In-Depth Information
Figure 6.4: Filter bank implementation of a one-dimensional discrete dyadic
wavelet transform decomposition and reconstruction for three levels of analysis.
H
s
(
ω
) denotes the complex conjugate of
H
s
(
ω
).
Fig. 6.4. Filters
F
(2
m
ω
) defined at level
m
+
1 (i.e., filters applied at wavelet scale
2
m
) are constructed by inserting 2
m
−
1 zeros between subsequent filter coeffi-
cients from level 1 (
F
(
ω
)). Noninteger shifts at level 1 are rounded to the nearest
integer. This implementation design is called “
algorithme a trous
” [17, 18] and
has a complexity that increases linearly with the number of analysis levels.
In image processing applications, we often deal with two, three, or even
higher dimensional data. Extension of the framework to higher dimension is
quite straightforward. Multidimensional wavelet bases can be constructed with
tensor products of separable basis functions defined along each dimension.
In that context, an
N
-dimensional discrete dyadic wavelet transform with
M
analysis levels is represented as a set of wavelet coefficients:
S
M
s
,
{
W
m
s
,
W
m
s
,...,
W
m
s
}
m
=
[
I
,
M
]
,
(6.17)
where
W
m
s
=
s
,ψ
k
m
represents the detailed information along the
k
th coordi-
nate at scale
m
. The wavelet basis is dilated and translated from a set of separable
wavelet functions
ψ
k
,
k
=
1
,...,
N
,
for example in 3D:
k
x
−
n
1
2
m
1
2
3
m
/
2
ψ
y
−
n
2
2
m
z
−
n
3
2
m
m
,
n
1
,
n
2
,
n
3
(
x
,
y
,
z
)
=
k
=
1
,
2
,
3
.
(6.18)
ψ
,
,
,
