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Unlocking the Power of Wavelets with Bi-orthogonal Extensions

Learn about the importance of symmetry in wavelets and how to construct bi-orthogonal families for complete orthonormal wavelet sets on intervals. Explore the challenges of filter coefficients, Vetterli conditions, and perfect reconstruction in sub-band coding. Delve into the world of Box, Haar, Daub4 functions, and discover the magic of Pythagoras in wavelet analysis.

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Unlocking the Power of Wavelets with Bi-orthogonal Extensions

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  1. Extensions 1.Bi-orthogonal wavelets 2.Wavelets on an interval 3.Wavelets on the plane 4. Wavelets on curves and surfaces 5.Lifting

  2. Deficiency 1: lack of symmetry Box: symmetric about Haar: symmetric about Theorem: (D) Haar family compactly supported completeorthonormal wavelet family with odd symmetry.

  3. Why symmetry? data sequences finite length 1. Pad with zeros 2. Periodic wavelets 3. Extend by symmetry Solution: (i)construct symmetric scaling and wavelet functions Penalty:can’t have orthonormal families So: biorthogonal families Solution: (ii) construct complete orthonormal wavelet families on intervals

  4. Deficiency 2: algebraic complications filter coefficients have to satisfy Coefficients found by taking square roots Solution? more general Vetterli conditions , Two filters:

  5. Filter banks, Sub-band coding again Vetterli conditions needed for perfect reconstruction and Bi-orthogonality.

  6. Deficiency 3: unfamiliarity Box, Haar , Tent functions familiar, Daub4 not so. Consider tent function centered at . But NOT orthonormal. Know filter: Frequency Response Function: But what about and ?

  7. Pythagoras again! also Factor, use trig identities: Set:

  8. Pythagoras again! and , as before so that , Vetterli conditions satisfied. Run recursive construction as before: where: ,

  9. Degree 1 case tent function,

  10. Fractal Scaling function Degree 1

  11. Mexican Hat function Degree 1

  12. Fractal Mexican Hat Degree 1

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