Biomedical Engineering Reference
In-Depth Information
which is the reconstruction part. The decomposition is even easier: We need
merely use the formulas for
a
n
and
b
n
to find
∞
∞
f
(
x
)
k
c
k
√
2
φ
(2
x
−
2
n
−
k
)
dx
a
0
,
n
=
f
(
x
)
φ
(
x
−
n
)
dx
=
(5.67)
−∞
−∞
=
c
k
a
1
,
2
n
+
k
=
a
1
,
k
c
k
−
2
n
,
k
k
a
1
,
k
(
−
1)
k
−
1
c
1
−
k
+
2
n
.
b
0
,
n
=
k
This works at each scale to give us the tree algorithm for decomposition
(5.67),
b
m
−
1
,
n
b
0
,
n
···−→
a
m
,
n
−→
a
m
−
1
,
n
−→ · · · −→
a
1
,
n
−→
a
0
,
n
−→···
and for reconstruction (5.66),
b
0
,
n
b
1
,
n
b
m
−
1
,
n
···−→
a
0
,
n
−→
a
1
,
n
−→ · · · −→
a
m
−
1
,
n
−→
a
m
,
n
−→···
.
Thus we need calculate the coefficients from the function
f
(
t
) only once at
the finest scale of interest. Then we work down to successively coarser scales
by using this decomposition algorithm, with the error at each successive scale
corresponding to the wavelet coefficients. These algorithms are called Mallat
algorithms (see [39]).
5.4.1.2 2D Separable Wavelets
In order to represent an image using wavelet bases, we need to construct a
basis for
L
2
(
R
2
)
.
There are two different methods to do so. One way is based on
the multiresolution analysis in 2D space to construct 2D wavelet basis directly,
while another way is based on the tensor product of the 1D wavelets. The former
usually leads to a nonseparable basis, while the latter derives a separable basis.
Here we merely consider the separable basis, which is based on the separable
multiresolution analysis of
L
2
(
R
2
)
.
