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 H m (z):

9. FIR Filters For finite impulse response (FIR) filters, the determinate of H m is a pure delay, i.e., Let  =2 Let  =-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 h k (z) are defined over the continuous, closed interval [0,1] for k=0,1,..N-1 where N=2 n.

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  wisely, {  j,k (x)} can be made to span L 2 (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 V j is f(x)=0 Requirement 4: Any function can be represented with arbitrary precision.

24. Wavelet Functions

25. A wavelet function,  (x), together with its integer translates and binary scalings, spans the difference between any two adjacent scaling subspace, V j and V j+1.

26. Haar Wavelet Functions

27. Wavelet Series Expansion

28. Harr Wavelet Series Expansion of y=x 2

29. Discrete Wavelet Transform

30. The Continuous Wavelet Transform

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