Information Technology Reference
In-Depth Information
This also shows that the sum of the squares of the scaling coefficients is equal to
2. The same coefficients are used in reverse with alternate signs to produce the
associated wavelet equation
.x/ D
X
k
.1/
k
c
1k
.2x k/
(11.23)
This construction ensures that the wavelets and their corresponding scaling func-
tions are orthogonal. For wavelets of compact support, which have a finite number
of scaling coefficients N
k
the following wavelet function is defined
.x/ D
X
k
.1/
k
c
N
k
1k
.2x k/
(11.24)
This ordering of scaling coefficients used in the wavelet equation allows for our
wavelets and their corresponding scaling equations to have support over the same
interval Œ0;N
k1
. Often the reconfigured coefficients used for the wavelet function
are written more compactly as
b
k
D .1/
k
c
N
k
1k
(11.25)
where the sum of all coefficients b
k
is zero. Using this reordering of the coefficients
Eq. (
11.24
) can be written as
N
k
1
X
.x/ D
b
k
.2x k/
(11.26)
kD0
From the previous equations and examining the wavelet at scale index m C 1 one
can see that for arbitrary integer values of m the following holds
2
.mC1/=2
2
mC1
n
c
k
2t
22
m
2n k
D 2
m=2
2
1=2
X
k
1
(11.27)
which may be written more compactly as
p
2
X
k
1
mC1;n
.x/ D
c
k
m;2nCk
.x/
(11.28)
That is the scaling function at an arbitrary scale is composed of a sequence of
shifted functions at the next smaller scale each factored by their respective scaling
coefficients. Similarly, for the wavelet function one obtains
p
2
X
k
1
mC1;n
.x/ D
b
k
m;2nCk
.x/
(11.29)
