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work, except for two factors. We developed this expansion without using any of the machinery
of wavelets, in fact without explicitly using the idea of the frequency domain. All the devel-
opment was in the spatial domain using spatial concepts like prediction. Secondly, we have
used very simple (the simplest) predictor and update operators. By using more complicated
predictors and update operators, we can get more sophisticated wavelet implementations.
Let's use slightly more complicated prediction and update operators. Instead of using
single neighbor lets predict an odd-indexed value by using the the average of two even-indexed
neighbors:
x
e
,
k
+
x
e
,
k
+
1
2
d
k
=
x
o
,
k
−
x
2
k
+
x
2
k
+
2
2
=
x
2
k
+
1
−
Notice that if the input sequence corresponds to a linear or constant function, in other words
a polynomial of degree less than 2, the detail coefficients will all be zero.
For the corresponding update operator, let's use the weighted sumof the residual neighbors:
s
k
=
x
e
,
k
+
α(
d
k
−
1
+
d
k
)
To preserve the average value, we need to satisfy Equation (
94
):
k
s
k
=
x
e
,
k
+
α(
d
k
)
d
k
−
1
+
k
x
2
k
+
α
x
2
k
−
1
−
x
2
k
−
2
+
x
2
k
x
2
k
+
x
2
k
+
2
2
=
+
x
2
k
+
1
−
2
k
=
(
1
−
2
α)
x
2
k
+
2
α
x
2
k
+
1
k
k
Again, assuming that the sum of the even-indexed components and the odd-indexed compo-
nents are the same, we get
1
2
1
−
2
α
=
2
α
=
which implies that
α
equals 1
/
4. Thus,
x
e
,
k
+
(
d
k
−
1
+
d
k
)
s
k
=
4
This is an implementation of the Cohen-Daubechies-Feaveau (2,2) (CDF(2,2)) wavelet [
213
].
The numbers in parenthesis indicate the number of moments that are preserved in the update
step and the order of the input polynomial below which the detail coefficients will be zero. In
both of these examples we have used one prediction and one update step. In practice this is not
a restriction. We can use any number of update and prediction steps to implement more and
more sophisticated wavelet decompositions. An example that will be especially useful to us
in the next chapter is CDF(4,4) biorthogonal wavelets. This requires two prediction and two
