1. Entropy-constrained overcomplete-based coding of natural images André F. de Araujo, Maryam Daneshi, Ryan Peng Stanford University

2. 2 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Outline  Motivation  Overcomplete-based coding: overview  Entropy-constrained overcomplete-based coding  Experimental results  Conclusion  Future work

3. 3 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Motivation (1)  Study of new (and unusual) schemes for image compression  Recently, new methods have been developed using the overcomplete approach  Restricted scenarios for compression  Did not fully exploit this approach’s characteristics for compression

4. 4 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Motivation (2) Why? Sparsity on coefficients  better overall RD

5. 5 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Overcomplete coding: overview (1)  K > N implies:  Bases are not linearly independent  Example:  8x8 blocks: N = 64 basis functions are needed to span the space of all possible signals  Overcomplete basis could have K = 128  Two main tasks: 1. Sparse coding 2. Dictionary learning

6. 6 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Overcomplete coding: overview (2) 1. Sparse coding (“atom decomposition”)  Compute the representation coefficients x based on the signal y (given) and dictionary D (given)  overcomplete D  Infinite solutions  approxim.  Commonly used algorithms: Matching Pursuits (MP), Orthogonal Matching Pursuits (OMP)

7. 7 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Overcomplete coding: overview (3) Sparse coding (OMP)

8. 8 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Overcomplete coding: overview (4) 2. Dictionary learning  Two basic stages (analogy with K-means) i. Sparse coding stage: use a pursuit algorithm to compute x (OMP is usually employed) ii. Dictionary update stage: adopt a particular strategy for updating the dictionary  Convergence issues: as first stage does not guarantee best match, cost can increase and convergence cannot be assured

9. 9 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Overcomplete coding: overview (5) 2. Dictionary learning  Most relevant algorithms in the literature: K-SVD and MOD  Sparse coding stage is done in the same way  Codebook update stage is different:  MOD  Update entire dictionary using optimal adjustment for a given coefficients matrix  K-SVD  Update each basis one at a time using SVD formulation  Introduces change in dictionary and coefficients

10. 10 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Entropy-const. OC-based coding (1)

11. 11 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Entropy-const. OC-based coding (2)

12. 12 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Entropy-const. OC-based coding (3) RD-OMP

13. 13 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Entropy-const. OC-based coding (4)  EC Dictionary Learning – key ideas  Dictionary update strategy  K-SVD modifies dictionary and coefficients - reduction in Lagrangian cost is not assured.  We use MOD, which provides the optimal adjustment assuming fixed coefficients  Introduction of “Rate cost update” stage  Analogous to ECVQ algorithm for training data  Two pmfs must be updated: indexes and coefficients

14. 14 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Entropy-const. OC-based coding (5) EC-Dictionary Learning

15. 15 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Experiments (Setup)  Rate calculation: optimal codebook (entropy) for each subband  Test images: Lena, Boats, Harbour, Peppers  Training dictionary experiments  Training data: 18 Kodak downsampled (to 128x128) images (does not include images being coded)  Use of downsampled images to 128x128, due to very high computational complexity (for other experiments, higher resolutions were employed: 512x512, 256x256)

16. 16 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Experiments (Sparse Coding)  Comparison of Sparse coding methods

17. 17 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Experiments (Dict. learning)  Comparison of dictionary learning methods

18. 18 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Experiments (Compression schemes) (1)  1: Training and coding for the same image (dictionary is sent)  2: Training with a set of natural images and applying to other images

19. 19 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Experiments (Compression schemes) (2)

20. 20 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Experiments (Compression schemes) (3)

21. 21 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Conclusion  Improvement of sparse coding:  RD-OMP  Improvement of dictionary learning  Entropy-constrained overcomplete dictionary learning  Better overall performance compared to standard techniques

22. 22 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Future work  Extension of implementation to higher resolution images  Further investigation of trade-off between K and N  Evaluation against directional transforms  Low complexity implementation of the algorithms

23. 23 EE398A Project – Winter 2010/2011 Mar. 10, 2011 Experiments (trained dictionary)  K = 256