Entropy-constrained overcomplete-based coding of
Description: Entropy-constrained overcomplete-based coding of natural images André F. de Araujo, Maryam Daneshi, Ryan Peng Stanford University Outline Motivation Overcomplete-based coding: overview Entropy-constrained overcomplete-based coding
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slide1. Entropy-constrained overcomplete-based
coding of natural images André F. de Araujo, Maryam Daneshi, Ryan Peng
Stanford University<br>
slide2. Outline Motivation
Overcomplete-based coding: overview
Entropy-constrained overcomplete-based coding
Experimental results
Conclusion
Future work<br>
slide3. 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<br>
slide4. Motivation (2) Why? Sparsity on coefficients better overall RD<br>
slide5. 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:
Sparse coding
Dictionary learning<br>
slide6. Overcomplete coding: overview (2) 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)<br>
slide7. Overcomplete coding: overview (3) Sparse coding (OMP)<br>
slide8. Overcomplete coding: overview (4) Dictionary learning
Two basic stages (analogy with K-means)
Sparse coding stage: use a pursuit algorithm to compute x (OMP is usually employed)
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<br>
slide9. Overcomplete coding: overview (5) 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<br>
slide10. Entropy-const. OC-based coding (1)<br>
slide11. Entropy-const. OC-based coding (2)<br>
slide12. Entropy-const. OC-based coding (3) RD-OMP<br>
slide13. 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<br>
slide14. Entropy-const. OC-based coding (5) EC-Dictionary Learning<br>
slide15. 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)<br>
slide16. Experiments (Sparse Coding) Comparison of Sparse coding methods<br>
slide17. Experiments (Dict. learning) Comparison of dictionary learning methods<br>
slide18. 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<br>
slide19. Experiments (Compression schemes) (2)<br>
slide20. Experiments (Compression schemes) (3)<br>
slide21. Conclusion Improvement of sparse coding:
RD-OMP
Improvement of dictionary learning
Entropy-constrained overcomplete dictionary learning
Better overall performance compared to standard techniques<br>
slide22. 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<br>
slide23. Experiments (trained dictionary) K = 256<br>
coding of natural images André F. de Araujo, Maryam Daneshi, Ryan Peng
Stanford University<br>
slide2. Outline Motivation
Overcomplete-based coding: overview
Entropy-constrained overcomplete-based coding
Experimental results
Conclusion
Future work<br>
slide3. 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<br>
slide4. Motivation (2) Why? Sparsity on coefficients better overall RD<br>
slide5. 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:
Sparse coding
Dictionary learning<br>
slide6. Overcomplete coding: overview (2) 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)<br>
slide7. Overcomplete coding: overview (3) Sparse coding (OMP)<br>
slide8. Overcomplete coding: overview (4) Dictionary learning
Two basic stages (analogy with K-means)
Sparse coding stage: use a pursuit algorithm to compute x (OMP is usually employed)
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<br>
slide9. Overcomplete coding: overview (5) 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<br>
slide10. Entropy-const. OC-based coding (1)<br>
slide11. Entropy-const. OC-based coding (2)<br>
slide12. Entropy-const. OC-based coding (3) RD-OMP<br>
slide13. 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<br>
slide14. Entropy-const. OC-based coding (5) EC-Dictionary Learning<br>
slide15. 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)<br>
slide16. Experiments (Sparse Coding) Comparison of Sparse coding methods<br>
slide17. Experiments (Dict. learning) Comparison of dictionary learning methods<br>
slide18. 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<br>
slide19. Experiments (Compression schemes) (2)<br>
slide20. Experiments (Compression schemes) (3)<br>
slide21. Conclusion Improvement of sparse coding:
RD-OMP
Improvement of dictionary learning
Entropy-constrained overcomplete dictionary learning
Better overall performance compared to standard techniques<br>
slide22. 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<br>
slide23. Experiments (trained dictionary) K = 256<br>