Distributed Compression For Still Images

1. Distributed CompressionFor Still Images Kivanc Ozonat Distributed Compression For Still Images

2. Introduction • Description of the Problem • Related Concepts from Information Theory • Application of Bit-Plane Encoding as a Possible Solution Strategy • Proposed Solution: Using Transform Coding - Basic Scheme - Relation to the Information Theory Concepts Distributed Compression For Still Images

3. Problem Description • Given two still images, a noisy version, X, at the decoder and the original, Y, at the encoder, how to transmit Y with the best coding efficiency? • No communication of X and Y at the encoder Encoder Decoder Y X Distributed Compression For Still Images

4. Information Theory Background • Slepian-Wolf : Given the following scheme, (X,Y) (X,Y) R1 Encode X Encode Y R2 Distributed Compression For Still Images

5. Information Theory Background • Can transmit X and Y, if: - R1 > H(X|Y) , R2 > H(Y|X), and - R1+ R2 > H(X,Y). R2 H(Y) H(Y|X) R1 H(X|Y) H(X) Distributed Compression For Still Images

6. Information Theory Background • Our problem is a special case of this: R2 H(Y) H(Y|X) H(Y|X) R1 H(X|Y) H(X) H(X) Distributed Compression For Still Images

7. General Solution Strategy • Form cosets with 3 requirements: - Members of the same coset should be maximally separated. - Members of the same coset should have the same (or very close) probabilities of occurrence. - Coset construction should be practically implementable. Distributed Compression For Still Images

8. Underlying Approach • Use the Idea of Jointly Typical Sets: - Encode a long sequence (length n) of i.i.d. sources together, and form the typical set. - As n gets large, the typical set contains almost all of the probability of occurrence. - Further, the typical set has its members uniformly distributed. Distributed Compression For Still Images

9. Underlying Approach • The typical set contains most of the probability of occurrence, but it has only power (of 2) nH elements. Typical Set Distributed Compression For Still Images

10. Underlying Approach • Given i.i.d. X and i.i.d. Y, can form long sequences to get the typical X and typical Y sequences. • Then, there are power (of 2) H(X,Y) jointly typical sequences. Typical Y Typical X Distributed Compression For Still Images

11. A Possible Scheme? • Use bit-plane encoding (followed by gray coding) to divide the image at the encoder into bit-planes. • Exploit the correlation between the adjacent pixels through the upper bit planes. • The lower bit planes contain i.i.d. (or almost i.i.d) distribution of 0’s and 1’s. Distributed Compression For Still Images

12. A Possible Scheme? Example: A lower bit-plane, with i.i.d 0’s and 1’s: 0.7 0 0 0.3 0.3 1 1 0.7 Distributed Compression For Still Images

13. A Possible Scheme? • Given X , the noisy version, the jointly typical set is (X,Y) such that X is as given, and Y is the set with a Hamming distance of (0.3)n from X. • As n increases, Pr(X, Y=(X-0.3n)) will approach 1, with each (X,Y) pair having the same distribution. • Hence, can perform an efficient coset construction Distributed Compression For Still Images

14. A Possible Scheme? • Problems: • The lower bit-planes, which are of interest, have error probabilities of close to 0.5, even with moderate noise variances. - Cannot compete with transform domain methods in terms of bit rates. Distributed Compression For Still Images

15. Proposed Solution • Using transform coding is better because: - Energy compaction, resulting in lower bit rates for PSNR’s of around 38-40 dB. - The addition / averaging process involved in transform coding reduces the effect of noise through the central limit theorem. - The coded sequences of coefficients are de-correlated to a significant extent. Distributed Compression For Still Images

16. Proposed Solution • Schematically, Modified Huffman Coder Y DCT Zonal Coder Quantizer Coset Constructer Bit-Plane Encoder Distributed Compression For Still Images

17. Proposed Solution • The coefficients can be grouped in pairs with almost equal probabilities of occurrence. (because they are Laplacian) • One member from each pair is selected and Huffman-coded. • The other member of the pair is to have exactly the reverse (0’s and 1’s switched) code. • The coded coefficients are placed in bit-planes. Distributed Compression For Still Images

18. Proposed Solution • 1, 2, 3, 4, 5 • 01, 10, 11, 000, 111 (w.p. 0.125, 0.125, 0.10 , 0.075, 0.075) • -1, -2, -3, -4, -5 • 10, 01, 00, 111, 000 (w.p. 0.125, 0.125, 0.10 , 0.075, 0.075) Assume: need to code 1,-4,-2, 2, -3,-1,5,2,3 • 100100111 • 010000100 • 010101111 • 111000101 Distributed Compression For Still Images

19. Proposed Solution • Advantages: - Equal likelihood of 0’s and 1’s. This makes the use of the error coding simple. - Essentially Huffman coding. Not a significant increase, if the upper bit-planes are run-length coded. - Maximal distribution of cosets possible, due to the uniformity and equality of probabilities. Distributed Compression For Still Images

20. Conclusions • Asymptotic Equipartition Property is essential in forming the typical sets. • Having equal probabilities of occurrences make the error-coding simpler. • Decorrelation is maintained through DCT. • The Laplacian Distribution of the DCT coefficients is important in getting equally probable pairs. Distributed Compression For Still Images