Lifting and Biorthogonality

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Lifting and Biorthogonality

• Ref SIGGRAPH 95

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Projection Operator

• Def The approximated function in the subspace is
  obtained by the projection operator
• As j?, the approximation gets finer, and

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Projection Operator (cont)

• In general, it is hard to construct orthonormal
  scaling functions
• In the more general biorthogonal settings,

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Ex Linear Interpolating
biorthogonal!
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Ex Constant Average-Interpolating
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Think

• What does Pj1 look like in linear interpolating
  and constant AI?
• What does Pj look like in other lifting schemes?
  (cubic interpolating, quadratic AI, )

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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
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Ex MRA of Order 4

• as in the case of cubic predictor in lifting
• Pj can reproduce x0, x1, x2, and x3 (and any
  linear combination of them)

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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

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• From property of

j1 one level finer in MRA
This means The pth moment of finer and
coarsened approximations are the same.
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Summary

• If the dual order of MRA is
• Any polynomial of degree less than is
  reproducible by the dual projection operator
• Pj preserves up to moments
• If the order of MRA is N
• Any polynomial of degree less than N is
  reproducible by the projection operator Pj
• preserves up to N moments

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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
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Coarsening

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

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Ex Coarsening for Linear Interpolation
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Wavelets

• form a basis for the difference between two
  successive approximations
• Wavelet coefficients encode the difference of
  DOF between Pj and Pj1

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This implies
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MRA
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• Wj depends on
• how Pj is calculated from Pj1
• Hence, related to the dual scaling function

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Details
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Dual Wavelets

• To find the wavelet coefficients gj,m

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complement (refinement relation)
basis of
coeff. obtained by
basis of
complement (refinement relation)
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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

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Lifting changes

• Changes propagate as follows

Dual wavelet
Primal wavelet
Dual Scaling fn
Pj Computing Coarser rep.
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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)

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Inside Lifting (cont)

• U affects how sj-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

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