Lifting and Biorthogonality Ref: SIGGRAPH 95

Projection Operator Def: The approximated function in the subspace is obtained by the “projection operator” As j↑, the approximation gets finer, and

Projection Operator (cont) In general, it is hard to construct orthonormal scaling functions In the more general biorthogonal settings,

Ex: Linear Interpolating biorthogonal!

Ex: Constant Average-Interpolating j,k

Think … What does P j+1 look like in linear interpolating and constant AI? What does P j look like in other lifting schemes? (cubic interpolating, quadratic AI, …)

Polynomial Reproduction If the order of MRA is N, then any polynomial of degree less than N can be reproduced by the scaling functions That is, This is true for all j

Ex: MRA of Order 4 as in the case of cubic predictor in lifting … –P j can reproduce x 0, x 1, x 2, and x 3 (and any linear combination of them) …

Interchange the roles of primal and dual … Define the dual projection operator w.r.t. the dual scaling functions Dual order of MRA: –Any polynomial of degree less than is reproducible by the dual projection operator

From property of : j+1: one level finer in MRA This means: “The p th moment of finer and coarsened approximations are the same.” This means: “The p th moment of finer and coarsened approximations are the same.”

Summary If the dual order of MRA is –Any polynomial of degree less than is reproducible by the dual projection operator –P j preserves up to moments If the order of MRA is N –Any polynomial of degree less than N is reproducible by the projection operator P j – preserves up to N moments

Subdivision Assume The same function can be written in the finer space: The coefficients are related by subdivision: Recall “lifting-2.ppt”, p.16, 18

Coarsening On the other hand, to get the coarsened signal from finer ones: substitute the dual refinement relation into Recall

Ex: Coarsening for Linear Interpolation

Wavelets … form a basis for the difference between two successive approximations Wavelet coefficients: encode the difference of DOF between P j and P j+1 PjPj PjPj P j+1 P j+1 - P j

This implies … (primal) wavelet has vanishing moments

MRA VNVN V N-1 W N-1 V N-2 W N-2 V N-3 W N-3

W j depends on … –how P j is calculated from P j+1 Hence, related to the dual scaling function

Details

Dual Wavelets To find the wavelet coefficients  j,m

Primal Scaling Fns Dual Scaling Fns basis of coeff. obtained by Primal Wavelets Dual Projection Primal Projection Dual Wavelets basis of complement (refinement relation) complement (refinement relation)

Lifting (Basic Idea) Idea: taken an old wavelet (e.g., lazy wavelet) and build a new, more performant one by adding in scaling functions of the same level old wavelets scaling fns at level j scaling fns at level j+1 combine old wavelet with 2 scaling fns at level j to form new wavelet

Lifting changes … Changes propagate as follows: Primal wavelet Dual Scaling fn P j : Computing Coarser rep. Dual wavelet

Inside Lifting From above figure, we know P determines the primal scaling function (by sending in delta sequence) Different U determines different primal wavelets (make changes on top of the old wavelet)

Inside Lifting (cont) U affects how s j-1 to be computed (has to do with ). Scaling fns  are already set by P. ? From the same two-scale relations with (same ) Visualizing the dual scaling functions and wavelets by cascading