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Wavelet and multiresolution process

Wavelet and multiresolution process. Pei Wu 5.Nov 2012. Mathematical preliminaries: Some topology. Open set: any point A in the set must have a open ball O( r,A ) contained in the set. Closed set: complement of open set.

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Wavelet and multiresolution process

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  1. Wavelet and multiresolution process Pei Wu 5.Nov 2012

  2. Mathematical preliminaries: Some topology Open set: any point A in the set must have a open ball O(r,A) contained in the set. Closed set: complement of open set. Intersection of closed set is always closed. Union of open set is always open Compact: if we put infinite point in the set it must have infinity point “gather” around some point in the set. Complete: a “converge” sequence must converge at a point in the set.

  3. Mathematical preliminaries: Hilbert space • Hilbert space is a space… • linear • complete • with norm • with inner product • Example: Euclidean space, L2 space, …

  4. Mathematical preliminaries: orthonormal basis f,g is orthogonal iff <f,g>=0 f is normalized iff <f,f>=1 Orthonormal basis: e1, e2, e3,… s.t. a set of basis is called complete if

  5. equivalent condition for orthonormal A set of element {ei} is orthonormal if and only if: A orthonormal set induces isometric mapping between Hilbert space and l2.

  6. Motivation in context of Fourier transform we suppose the frequency spectrum is invariant across time: However in many cases we want:

  7. Example: Music

  8. Windowed Fourier Transform

  9. Analyze of Windowed Fourier transform A function cannot be localized in both time and frequency (uncertainty principle). High frequency resolution means low time resolving power.

  10. Trade-off between frequency resolution and time resolution

  11. Adaptive resolution Use big ruler to measure big thing, small ruler to measure small thing.

  12. Wavelet Use scale transform to construct ruler with different resolution.

  13. CWT(continuous wavelet transform)

  14. Proof (1)

  15. Proof (2)

  16. Discretizing CWT a,b take only discrete number: And we want them to be orthogonal:

  17. Example for wavelet (a)Meyer (b,c)Battle-Lemarie

  18. Example for wavelet (2) (d) Haar (e,f)Daubechies

  19. Constructing orthogonal wavelet Multiresolution analysis A series of linear subspace {Vi} that:

  20. Example

  21. From scaling function to wavelet Firstly we find a set of orthonormal basis in V0: hn would play important role in discrete analysis

  22. Example: Haar wavelet

  23. Relaxing orthogonal condition is linearly independent but not orthogonal. is orthonormal basis of V0

  24. Example: Battle-Lemarie Wavelet Use spline to get continuous function

  25. Meyer Wavelet: compact support

  26. Fast Wavelet transform Mallat algorithm : top-down Given c1 how can we get c0 and d0? Given c0 and d0 how to reconstruct c1 ?

  27. Mallat algorithm (2)

  28. Mallat algorithm (3):frequency domain perspect Subband coding

  29. Adaptive resolution

  30. 2D Wavelet Wavelet expansion of 2D function Basis for 2D function:

  31. Mallet algorithm

  32. Frequency Domain Decomposition

  33. Denoise using wavelet

  34. Wavelet packet We can carry on decomposition on high-frequency part Adaptive approach to decide decompose or not.

  35. Demo: finger-print image

  36. Demo: finger-print image

  37. Thank You!!

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