1. Wavelets and Multi-resolution Processing
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 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