Databases Reference
In-Depth Information
Example15.5.3:
Consider the case when
k
=
4. The three conditions give us the following three equations:
√
2
h
0
+
h
1
+
h
2
+
h
3
=
(84)
h
0
+
h
1
+
h
2
+
h
3
=
1
(85)
h
0
h
2
+
h
1
h
3
=
0
(86)
We have three equations and four unknowns; that is, we have one degree of freedom. We can
use this degree of freedom to impose further conditions on the solution. The solutions to these
equations include the Daubechies four-tap solution:
+
√
3
4
√
2
,
+
√
3
4
√
2
,
−
√
3
4
√
2
,
−
√
3
4
√
2
1
3
3
1
h
0
=
h
1
=
h
2
=
h
3
=
Given the close relationship between the scaling function and the wavelet, it seems reason-
able that we should be able to obtain the coefficients for the wavelet filter from the coefficients
of the scaling filter. In fact, if the wavelet function is orthogonal to the scaling function at the
same scale
φ(
t
−
k
)ψ(
t
−
m
)
dt
=
0
(87)
then
k
h
N
−
k
w
k
=±
(
−
1
)
(88)
and
h
k
w
n
−
2
k
=
(89)
0
k
Furthermore,
w
k
=
0
(90)
k
The proof of these relationships is somewhat involved [
212
].
15.5.2 Families of Wavelets
We have said that there is an infinite number of possible wavelets. Which one is best depends
on the application. In this section, we list different wavelets and their corresponding filters.
You are encouraged to experiment with these to find those best suited to your application.
The 4-tap, 12-tap, and 20-tap Daubechies filters are shown in Tables
15.1
,
15.2
,
15.3
.The
6-tap, 12-tap, and 18-tap Coiflet filters are shown in Tables
15.4
,
15.5
,
15.6
.
15.6 Biorthogonal Wavelets
We have focused on orthogonal wavelets in order to develop various concepts regardingwavelet
decomposition. However, in various image compression schemes, which will be the focus of
