Extensions

1
Extensions
1. Bi-orthogonal wavelets
2. Wavelets on an interval
3. Wavelets on the plane
4. Wavelets on curves and surfaces
5. Lifting
2
Deficiency 1 lack of symmetry
Box
symmetric about
Haar
symmetric about
Theorem (D)
Haar family
compactly supported complete orthonormal wavelet
family with odd symmetry.
3
Why symmetry?
data sequences finite length
1. Pad with zeros
2. Periodic wavelets
3. Extend by symmetry
Solution (i) construct symmetric scaling and
wavelet
functions
Penalty cant have orthonormal families
So biorthogonal families
Solution (ii) construct complete orthonormal

wavelet families on intervals
4
Deficiency 2 algebraic complications
filter coefficients have to satisfy
Coefficients found by taking square roots
Solution?
more general Vetterli conditions
,
Two filters
5
Filter banks, Sub-band coding again
Vetterli conditions needed for perfect
reconstruction
and Bi-orthogonality.
6
Deficiency 3 unfamiliarity
Box, Haar , Tent functions familiar, Daub 4 not
so.
Consider tent function centered at
.
But
NOT orthonormal.
Know filter
Frequency Response Function
But what about and ?
7
Pythagoras again!
also
Factor, use trig identities
Set
8
Pythagoras again!
and , as before so that
,
Vetterli conditions satisfied.
Run recursive construction as before
where
,
9
Degree 1 case
tent function,
10
Fractal Scaling function
Degree 1
11
Mexican Hat function
Degree 1
12
Fractal Mexican Hat
Degree 1