Databases Reference
In-Depth Information
That is, both the analysis and synthesis functions are carried out using the same basis set. In
this case of the biorthogonal expansion, analysis and synthesis are carried out by dual basis
sets. In the development above, the synthesis is carried out using the set
{
g
k
(
t
)
}
as in Equation
{˜
g
j
(
)
}
(
91
) while the analysis is carried out using the set
as in Equation (
92
).
In terms of wavelets, if we have a scaling function that satisfies the multiresolution analysis
equation
t
h
k
√
2
φ(
t
)
=
φ(
2
t
−
k
)
k
and a dual set that also satisfies the MRA
h
k
√
2
φ(
φ(
t
)
=
2
t
−
k
)
k
{
φ(
and the two sets
{
φ(
t
)
}
and
t
)
}
are orthogonal, that is
), φ(
φ(
t
t
−
k
)
=
δ(
k
)
then we can obtain analysis filters from one set and synthesis filters from the other [
212
].
Because we do not have to obtain all four filters from the same mother wavelet or scaling
function, we have much more freedom in picking wavelets to use. In particular, we can find
scaling functions and wavelets with linear phase implementations.
15.7 Lifting
We have discussed the idea of wavelet decomposition mainly in terms of the separation of
signals in the frequency domain. The input signal, through appropriate use of filters based on
wavelets, gets decomposed into a low-frequency signal and a high-frequency signal. The low-
and high-frequency signals are then downsampled, encoded, and transmitted. The received
signal is decoded and upsampled, and the low- and high-frequency signals are recovered
using low- and high-frequency reconstruction filters. An alternative way of looking at the
decomposition is to view it on spatial, or temporal, terms. Frequency can be thought of as
a way to measure how fast a signal changes either in space or in time, and as we saw in
Chapter 11, the slowly changing components of a signal can be extracted through the use of
prediction. Thus, high frequencies correspond to less predictable components of the signal and
low frequencies correspond to more predictable components of the signal where the prediction
is based on spatial or temporal correlation. This fact is utilized in an efficient implementation
of wavelets developed by Wim Sweldens called
lifting
[
213
].
We began Chapter 14 on subband coding with a simple motivating example where we
separated a sequence into its even-indexed and odd-indexed components. We return to that
example again using different terminology to motivate our discussion of lifting. Consider the
system shown in Figure
15.16
where the box labeled
Split
breaks the incoming sequence into
the even- and odd-indexed sequences. At the decoder, the box
Merge
performs the opposite
function. If the sequence is relatively slow changing, we can reduce the data rate by predicting
