Extensions 1 Biorthogonal wavelets 2 Wavelets on an

Extensions 1. Bi-orthogonal wavelets 2. Wavelets on an interval 3. Wavelets on the plane 4. Wavelets on curves and surfaces 5. Lifting

Deficiency 1: lack of symmetry Box: symmetric about Haar: symmetric about Theorem: (D) Haar family compactly supported complete orthonormal wavelet family with odd symmetry.

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: can’t have orthonormal families So: biorthogonal families Solution: (ii) construct complete orthonormal wavelet families on intervals

Deficiency 2: algebraic complications filter coefficients have to satisfy Coefficients found by taking square roots Solution? more general Vetterli conditions Two filters: ,

Filter banks, Sub-band coding again Vetterli conditions needed for perfect reconstruction and Bi-orthogonality.

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 ?

Pythagoras again! also Factor, use trig identities: Set:

Pythagoras again! and , as before so that , Vetterli conditions satisfied. Run recursive construction as before: where: ,

Degree 1 case tent function,

Fractal Scaling function Degree 1

Mexican Hat function Degree 1

Fractal Mexican Hat Degree 1