Wavelets and Multi-resolution Processing: Background, Image Pyramids, Z-Transform, FIR Filters, Bi-orthogonality, Haar T

1. Chapter 7 Wavelets and Multi-resolution Processing

2. Background

3. Image Pyramids Total number of elements in a P+1 level pyramid for P>0 is

4. Example

5. Subband Coding • An image is decomposed into a set of band-limited components, called subbands, which can be reassembled to reconstruct the original image without error.

6. Z-Transform • The Z-transform of sequence x(n) for n=0,1,2 is: • Down-sampling by a factor of 2: • Up-sampling by a factor of 2:

7. Z-Transform (cont’d) • If the sequence x(n) is down-sampled and then up-sampled to yield x^(n), then: • From Figure 7.4(a), we have:

8. Error-Free Reconstruction • Matrix expression • Analysis modulation matrix Hm(z):

9. FIR Filters • For finite impulse response (FIR) filters, the determinate of Hm is a pure delay, i.e., • Let a=2 • Let a=-2

10. Bi-orthogonality • Let P(z) be defined as: • Thus, • Taking inverse z-transform: • Or,

11. Bi-orthogonality (Cont’d) • It can be shown that: • Or, • Examples: Table 7.1

12. Table 7.1

13. 2-D Case

14. Daubechies Orthonormal Filters

15. Example 7.2

16. The Haar Transform • Oldest and simplest known orthonormal wavelets. • T=HFH where F: NXN image matrix, H: NxN transformation matrix. • Haar basis functions hk(z) are defined over the continuous, closed interval [0,1] for k=0,1,..N-1 where N=2n.

17. Haar Basis Functions

18. Example

19. Multiresolution Expansions • Multiresolution analysis (MRA) • A scaling function is used to create a series of approximations of a function or image, each differing by a factor of 2. • Additional functions, called wavelets, are used to encode the difference in information between adjacent approximations.

20. Series Expansions • A signal f(x) can be expressed as a linear combination of expansion functions: • Case 1: orthonormal basis: • Case 2: orthogonal basis: • Case 3: frame:

21. Scaling Functions • Consider the set of expansion functions composed of integer translations and binary scaling of the real, square-integrable function, ,i.e., • By choosing j wisely, {jj,k(x)} can be made to span L2(R)

22. Haar Scaling Function

23. MRA Requirements • Requirement 1: The scaling function is orthogonal to its integer translates. • Requirement 2:The subspaces spanned by the scaling function at low scales are nested within those spanned at higher resolutions. • Requirement 3:The only function that is common to all Vj is f(x)=0 • Requirement 4: Any function can be represented with arbitrary precision.

24. Wavelet Functions

25. Wavelet Functions • A wavelet function, y(x), together with its integer translates and binary scalings, spans the difference between any two adjacent scaling subspace, Vj and Vj+1.

26. Haar Wavelet Functions

27. Wavelet Series Expansion

28. Harr Wavelet Series Expansion of y=x2

29. Discrete Wavelet Transform

30. The Continuous Wavelet Transform

31. Misc. Topics • The Fast Wavelet Transform • Wavelet Transform in Two Dimensions • Wavelet Packets