1. Multiresolution Analysis (Section 7.1) CS474/674 – Prof. Bebis

2. Multiresolution Analysis Many signals or images contain information at different scales or levels of detail (e.g., people vs buildings) Analyzing information at the same scale will not be effective. Use windows of different size (i.e., vary scale)

3. Multiresolution Analysis (cont’d) Small size objects should be examined at a high resolution High resolution Low resolution Alternatively, use the same window size but analyze information at different resolutions: Large size objects should be examined at a low resolution

4. Multiresolution Analysis (cont’d) We will review two techniques for representing multiresolution information efficiently: –Pyramidal coding –Subband coding

5. high resolution j = J low resolution j=0 A collection of decreasing resolution images arranged in the shape of a pyramid. Image Pyramid (low scale) (high scale) Example: if N=256, there will be 8+1=9 levels

6. high resolution j = J low resolution j=0 A collection of decreasing resolution images arranged in the shape of a pyramid. Image Pyramid (low scale) (high scale) Example: if N=256, there will be 8+1=9 levels

7. A collection of decreasing resolution images arranged in the shape of a pyramid. Image Pyramid Example: if N=256, there will be 8+1=9 levels

8. high resolution j = J low resolution j=0 A collection of decreasing resolution images arranged in the shape of a pyramid. Image Pyramid (low scale) (high scale) If N=256, the pyramid will have 8+1=9 levels

9. Pyramidal coding Two pyramids: approximation and prediction residual averaging  mean pyramid Gaussian  Gaussian pyramid no filter  subsampling pyramid nearest neighbor biliner bicubic (details)

10. Pyramidal coding (cont’d) Approximation pyramid (based on Gaussian filter) Prediction residual pyramid (based on bilinear interpolation) Note: the last level is the same as that of the approximation pyramid

11. Pyramidal coding (cont’d) In the absence of quantization errors, the approximation pyramid can be re-constructed from the prediction residual pyramid.

12. Pyramidal coding (cont’d) Just add the “details” or “residual error” back! (details)

13. Pyramidal coding (cont’d) In the absence of quantization errors, the approximation pyramid can be re-constructed from the prediction residual pyramid. We only need to keep the prediction residual pyramid only! - More efficient representation

14. Subband coding Decompose an image (or signal) into different frequency bands (analysis step). Decomposition is performed so that the subbands can be re-assembled to reconstruct the original image without error (synthesis step) Need to choose appropriate filters (i.e., “filter bank”)

15. Subband coding – 1D Example f lp (n): approximation of f(n) f hp (n): detail of f(n) 2-band decomposition low pass high pass

16. Subband coding (cont’d) It can be shown that for perfect reconstruction, the synthesis filters (g 0 (n) and g 1 (n)) must be modulated versions of the analysis filters (h 0 (n), h 1 (n)): or original modulated

17. Subband coding (cont’d) Of special interest, are filters satisfying orthonormality conditions: An orthonormal filter bank can be designed from one prototype filter: (non-orthonormal filters require two prototype filters)

18. Subband coding (cont’d) Example: orthonornal filters prototype filter

19. Subband coding – 2D Example 4-band decomposition (using separable filters) approximation vertical detail horizontal detail diagonal detail LL LH HL HH low pass high pass

20. Subband coding (cont’d) approximation vertical detail horizontal detail diagonal detail