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Biorthogonal Wavelets

Biorthogonal Wavelets. Ref: Rao & Bopardikar, Ch.4 Jyun-Ming Chen Spring 2001. Ortho normal bases further simplify the computation. Why is orthogonality useful. Ortho v. Non-Ortho Basis. Sum of projection vectors !?. Dual Bases. Dual Basis. a 1 -a 2 and b 1 -b 2 are biorthogonal.

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Biorthogonal Wavelets

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  1. Biorthogonal Wavelets Ref: Rao & Bopardikar, Ch.4 Jyun-Ming Chen Spring 2001

  2. Orthonormal bases further simplify the computation Why is orthogonality useful

  3. Ortho v. Non-Ortho Basis Sum of projection vectors !?

  4. Dual Bases Dual Basis a1-a2 and b1-b2 are biorthogonal

  5. Dual Basis (cont) Verify duality ! • Dual basis may generate different spaces • Here: a1-a2 and b1-b2 generate two different 2D subspaces in Euclidean 3space. • Semiorthogonal: • For dual basis that generates the same subspace • Orthogonal: • Primal and dual are the same bases

  6. Extend to Function Space • MRA types: • orthogonal, semiorthogonal, biorthognal • Extend the concept to using biorthogonal MRA • More flexible design • Lifting scheme: a general design method for biorthogonal wavelets

  7. Alternative Wavelets: Biorthogonal Wavelets Proposed by Cohen (1992)

  8. Decomposition and reconstruction filters are FIR and have the same length Generally do not have closed-form expressions Usually not symmetric (linear phase) Haar wavelet is the only real-valued wavelet that is compactly supported, symmetric and orthogonal Higher-order filters (with more coefficients) have poor time-frequency localization Desired property: perfect reconstructionFIRsymmetric (linear-phase) filters Not available in orthogonal bases Characteristics of Orthogonal Basis

  9. delegate the responsibilities of analysis and synthesis to two different functions (in the biorthogonal case) as opposed to a single function in the orthonormal case more design freedom compactly supported symmetric analyzing and synthesis wavelets and scaling functions The Need for Biorthogonal Basis

  10. Biorthogonal Scaling Functions • Two sequences serve as impulse response of FIR filters • Two sets of scaling functions generate subspaces respectively • The basis are orthogonal; the two MRAs are said to be biorthogonal to each other dual

  11. Dual MRA (cont) • Basis of • Translated copy of appropriate dilation of

  12. Function approximation in subspaces Coarser approx Finer approx

  13. Relation between Finer and Coarser Coefficients

  14. Dual • Two sets of wavelets generate subspaces respectively • The basis are orthogonal; the two MRAs are said to be biorthogonal to each other Biorthogonal Wavelets Require:

  15. Two-scale relations of wavelet: primal and dual

  16. Function Projection m=2n+l

  17. Function Reconstruction

  18. Filter Bank

  19. VN VN-1 WN-1 WN-2 VN-2 VN-3 WN-3 Primal and Dual MRA (biorthogonal)

  20. Filter Relations (between primal and dual) Similarly,

  21. Filter Relations (cont) Similarly,

  22. Design of Biorthogonal Wavelets • because there is quite a bit of freedom in designing the biorthogonal wavelets, there are no set steps in the design procedure. … • Lifting (Sweldens 94): a scheme for custom-design biorthogonal wavelets

  23. Common property: Differences: if orthogonal: scaling functions (and wavelets) of the same level are orthogonal to each other If semiorthogonal, wavelets of different levels are orthogonal (from nested space) VN VN-1 WN-1 WN-2 VN-2 VN-3 WN-3 Special Cases: orthogonal and semiorthogonal Dual and primal are the same

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