Scaling
functions
Fix filter
‘or connect the dots’
no restrictions yet:
FUND. DEFN: Scaling Function
relates
at two levels of resolution.
Basic condition:
Examples so
far:
Box:
Tent centered at
:
Daubechies D4: does there exist
?
Fractal
example:
Dyadic
rationals:
determined at dyadic rationals
:
Convolution on integers? Powers of 2?
KNOW
Construct
ALL , THEN KNOW
on all
ALL
as limiting fixed point!
Iterative
process:
with limit
Construct sequence functions
such that
Then
What about convergence? Pointwise, in Energy?
Pointwise: start with Tent function
In energy: start with Box function
Getting
started:
Tent function centered at origin:
Basic idea: set
for suitable
Filter conditions:
Need
so that
Conditions on
:
Solve using Fourier Transforms as usual.
in
Fourier
Transforms:
Set
Then
So:
Up-sampling
again!
Recall
Crucial results:
where in z-transform notation:
Use these to compute
.
Connect the dots!
Daub-4
Depths: 1, 2, 4, 6
Cascade Algorithm:
convergence in energy
Start with box function: can exploit orthonormality.
with
as before, but no Vetterli condition yet. So
Orthonormality:
Case: k = 0
Can we recognize sequence:
?
Finally Vetterli!
Consider first:
Crucial identification:
Fourier transform:
Finally Vetterli!
When
we deduce that
So
, hence
,
ORTHONORMAL FAMILY for each k.
In the limit!
When
in energy, then
so Vetterli ensures
orthonormal family in
.
Finally wavelets:
Fix FIR filter
Assume convergence in energy and Vetterli.
Set
define wavelet by
compactly supported if
By same argument as for
d to identify
compactly supported.
:
.
More results for
wavelets:
Recall
,
so
By Vetterli yet again:
so
Thus
,
orthonormal family.
Still more results:
By same argument yet again:
But by Fourier transforms yet again:
where remember,
.
Thus:
all .
Main Theorem Part 1:
FIR filter
If
then
is a continuous function with
derivatives.
Main Theorem Part 2:
Suppose
also satisfies Vetterli condition.
Define wavelet:
.
Then:
1.
,
orthonormal families,
,
2.
3.
complete orthonormal family in
.
Main Theorem applied to Daub-4:
so
hence Daub-4
continuous,
not quite differentiable