Digital Signal Processing Reference
In-Depth Information
Figure 3.15. Analysis scaling and wavelet function in the Fourier domain:
(left) scaling function Fourier transform
ˆ
φ
; (right) wavelet function Fourier
transform
ˆ
ψ
.
If the wavelet function is the difference between two resolutions, that is,
ˆ
ˆ
ˆ
ψ
(2
ν
)
=
φ
(
ν
)
−
φ
(2
ν
)
,
(3.42)
which corresponds to
h
(
g
(
ν
)
=
1
−
ν
)
,
(3.43)
then the Fourier transform of the wavelet coefficients ˆ
w
j
(
ν
) can be computed as
c
j
−
1
(
).
In Fig. 3.15, the Fourier transform of the scaling function derived from the B
3
-
spline (see equation (3.40)) and the corresponding wavelet function Fourier trans-
form (equation (3.42)) are plotted.
ν
)
−
c
j
(
ν
3.7.2.2 Reconstruction
If the wavelet function is chosen as in equation (3.42), a trivial reconstruction from
W ={
w
1
,...,w
J
,
c
J
}
is given by
J
ν
=
ν
+
w
ν
.
c
0
(
)
c
J
(
)
ˆ
j
(
)
(3.44)
j
=
1
But this is a particular case, and other alternative wavelet functions can be chosen.
The reconstruction can be made scale by scale, starting from the lowest resolution
and designing appropriate dual-synthesis filters (
h
g
). From equations (3.33) and
(3.37), we seek the scaling coefficients
c
j
knowing
c
j
+
1
,
,
w
j
+
1
,
h
, and
g
. This can be
cast as a weighted least-squares problem written in the Fourier domain as
)
c
j
+
1
(
)
2
p
h
(2
j
h
∗
(2
j
min
c
j
(
ν
ν
)
−
ν
)
c
j
(
ν
ν
)
(3.45)
)
ˆ
)
2
p
g
(2
j
g
∗
(2
j
+
ν
w
j
+
1
(
ν
)
−
ν
)
c
j
(
ν
