1 / 31

Chapter 7

Chapter 7. Wavelets and Multi-resolution Processing. Background. Image Pyramids. Total number of elements in a P+1 level pyramid for P>0 is. Example. Subband Coding.

ryo
Download Presentation

Chapter 7

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related