The Lifting Scheme: a customdesign construction of biorthogonal wavelets

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The Lifting Schemea custom-design construction
of biorthogonal wavelets

• Sweldens95, Sweldens 98
• (appeared in SIAM Journal on Mathematical
  Analysis)

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Relations of Biorthogonal Filters
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Biorthogonal Scaling Functions and Wavelets
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Wavelet Transform(in operator notation)
Filter operators are matrices encoded with filter
coefficients with proper dimensions
Note that up/down-sampling is absorbed into the
filter operators
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Operator Notation
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Relations on Filter Operators
Biorthogonality
Write in matrix form
Exact Reconstruction
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Theorem 8 (Lifting)

• Take an initial set of biorthogonal filter
  operators
• A new set of biorthogonal filter operators can be
  found as
• Scaling functions and H and untouched

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Proof of Biorthogonality
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Choice of S

• Choose S to increase the number of vanishing
  moments of the wavelets
• Or, choose S so that the wavelet resembles a
  particular shape
• This has important applications in automated
  target recognition and medical imaging

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Corollary 6.
Same thing expressed in frequency domain

• Take an initial set of finite biorthogonal
  filters
• Then a new set of finite biorthogonal filters can
  be found as
• where s(w) is a trigonometric polynomial

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Details
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Theorem 7 (Lifting scheme)

• Take an initial set of biorthogonal scaling
  functions and wavelets
• Then a new set , which is formally
  biorthognal can be found as
• where the coefficients sk can be freely chosen.

Same thing expressed in indexed notation
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Dual Lifting

• Now leave dual scaling function and and G
  filters untouched

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Fast Lifted Wavelet Transform

• Basic Idea never explicitly form the new
  filters, but only work with the old filter, which
  can be trivial, and the S filter.

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Before Lifting
Forward Transform
Inverse Transform
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Examples

• Interpolating Wavelet Transform
• Biorthogonal Haar Transform

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The Lazy Wavelet

• Subsampling operators E (even) and D (odd)

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Interpolating Scaling Functions and Wavelets

• Interpolating filter always pass through the
  data points
• Can always take Dirac function as a formal dual

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Theorem 15

• The set of filters resulting from interpolating
  scaling functions, and Diracs as their formal
  dual, can be seen as a dual lifting of the Lazy
  wavelet.

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(No Transcript)
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Algorithm of Interpolating Wavelet Transform
(indexed form)
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Example Improved Haar

• Increase vanishing moments of the wavelets from 1
  to 2
• We have

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Verify Biorthogonality
Details
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Improved Haar (cont)
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g(0) g(0) 0
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Verify Biorthogonality
Details