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

Why is orthogonality useful Orthonormal bases further simplify the computation

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

Dual Basis Dual Bases a 1 -a 2 and b 1 -b 2 are biorthogonal

Dual Basis (cont) Dual basis may generate different spaces –Here: a 1 -a 2 and b 1 -b 2 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 Verify duality !

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

Alternative Wavelets: Biorthogonal Wavelets Proposed by Cohen (1992)

Characteristics of Orthogonal Basis 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 reconstruction FIR symmetric (linear-phase) filters –Not available in orthogonal bases

The Need for Biorthogonal Basis 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

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

Dual MRA (cont) Basis of –Translated copy of appropriate dilation of

Function approximation in subspaces Coarser approx Finer approx

Relation between Finer and Coarser Coefficients

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

Two-scale relations of wavelet: primal and dual

Function Projection m=2n+l

Function Reconstruction

Filter Bank

Primal and Dual MRA (biorthogonal) VNVN V N-1 W N-1 V N-2 W N-2 V N-3 W N-3

Filter Relations (between primal and dual) Similarly,

Filter Relations (cont) Similarly,

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

Special Cases: orthogonal and semiorthogonal 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) VNVN V N-1 W N-1 V N-2 W N-2 V N-3 W N-3 Dual and primal are the same