Polynomials
P
N
x
P
N
x
have an interesting scaling property that if we
scale the domain variable xx by a
real number MM, the resulting
function is still a polynomial with the same degree. A
polynomial of degree NN over real
numbers is defined as follows:
P
N
x=∑k=0N
a
k
xk
P
N
x
k
0
N
a
k
x
k
(1)
The coefficients
a
k
a
k
of the monomial
xk
x
k
are real for real polynomials. In general they could
be complex numbers and
P
N
x
P
N
x
can be defined over the complex domain.
Consider
P
N
xM=fx
P
N
x
M
f
x
, we find
fx=∑k=0N
a
k
xMk=∑k=0N
a
k
Mkxk=∑k=0N
b
k
xk
f
x
k
0
N
a
k
x
M
k
k
0
N
a
k
M
k
x
k
k
0
N
b
k
x
k
(2)
So
fx
f
x
is also a polynomial over
xx with degree
NN.
We know that splines
are smoothly connected pieces of polynomials. Let
φx
φ
x
be a spline.
φx
φ
x
is a piecewise polynomial over integer intervals.
Dilate
φx
φ
x
by MM.
φxM
φ
x
M
is also a piecewise polynomial over integer
intervals with same degree. So it can be represented by using
B-Spline Bases.
In particular integer dilates of B-splines can be represented
by the unscaled B-spline bases themselves which gives us a
scaling equation for
B-Splines.
Consider shifted causal B-splines
φnx=
β
n
x-n+12
φ
x
n
β
n
x
n
1
2
. These are obtained by the (
n+1
n
1
)-fold convolution of
φ
0
x
φ
0
x
, the Haar scaling function.
φ
0
xM=∑k=0M-1
φ
0
x-k=∑k=0M-1
h
M
0
k
φ
0
x-k
φ
0
x
M
k
0
M
1
φ
0
x
k
k
0
M
1
h
M
0
k
φ
0
x
k
(3)
where
h
M
0
k
h
M
0
k
is the filter with z-transform
H
M
0
z=∑k=0M-1z-k
H
M
0
z
k
0
M
1
z
k
(discrete pulse of length
MM).
Convolve
φ
0
xM
φ
0
x
M
with itself
n+1
n
1
times. This gives us the following result:
φ
n
xM=∑k=0k∈ℤM-1
h
M
n
k
φ
n
x-k
φ
n
x
M
k
0
M
1
k
h
M
n
k
φ
n
x
k
(4)
where
H
M
n
z=1Mn
H
M
0
n+1zn1=1Mn∑k=0M-1z-k
H
M
n
z
1
M
n
H
M
0
z
n
1
1
M
n
k
0
M
1
z
k
(5)
Thus we have an
MM-scale
relation for all integers
MM.
If
M=2
M
2
,
H
2
n
z
H
2
n
z
is a binomial filter whose coefficients form the Pascal's triangle.
We saw that shifted B-splines
φ
n
x=
β
n
x-n+12
φ
n
x
β
n
x
n
1
2
are scaling functions.
We can form a nested set of subspaces
V
j
V
j
, such that
∀j,
V
j
⊂
V
j
+
1
:
V
j
=
Span
k
φ2jt-k
j
V
j
V
j
+
1
V
j
Span
k
φ
2
j
t
k
(6)
And they form a
multiresolution
analysis system that is dense in
L
2
L
2
. How do we find the spline wavelets?
φ
n
x-k
φ
n
x
k
are not orthogonal so we need to find either
biorthogonal or semi-orthogonal
wavelets. The biorthogonal spline wavelets are compact and
give FIR filters for the wavelet
filter bank. The semi-orthogonal spline
wavelets generate IIR filter bank structures. There are also
other spline wavelets such as Stromberg's one sided orthogonal
splines and the Battle-Lemarie Wavelets.
These are the wavelets obtained using biorthogonality of the
filters on synthesis and analysis sides of the filter
bank. For further details refer to BIORTHOGONAL WAVELETS.
-
φ
1
x=
β
1
x-1
φ
1
x
β
1
x
1
, the Hat function gives the 5/3 filter bank.
-
Hz=1+z-122
H
z
1
z
1
2
2
is the scaling filter for
φ
1
x
φ
1
x
.
-
The biorthogonal scaling filter is
H
^
z=1+z-122-1+4z-1+z-22
H
^
z
1
z
1
2
2
1
4
z
1
z
2
2
.
The scaling filters and wavelets generated are plotted in
Figure 1.
Basis
φt-k
φ
t
k
for
V
0
V
0
are non-orthogonal. We impose wavelets
wt-k
w
t
k
to be orthogonal to
φt
φ
t
.
V
0
⊥
W
0
⊥
V
0
W
0
and
V
0
⊕
W
0
V
0
W
0
as in orthogonal scaling functions.
wt-k
w
t
k
need not be an orthogonal set. The analysis
filters are IIR. The high pass synthesis filter
Gz
G
z
is given by
Gz=-z1-2NH-z-1A-z-1
G
z
z
1
2
N
H
z
A
z
(7)
H
~
z
H
~
z
and
G
~
z
G
~
z
are the low pass and high pass analysis filters
respectively.
H
~
z=2z-1Hz-1Az-1Az-2
H
~
z
2
z
H
z
A
z
A
z
2
(8)
G
~
z=2H-zAz-2
G
~
z
2
H
z
A
z
2
(9)
The
Az-2
A
z
2
in the denominator makes the analysis filters IIR, where
Az
A
z
is the z-transform of the sequence,
ak=<φt,φt-k>
a
k
φ
t
φ
t
k
.
