Distributed Compression For Still Images Kivanc Ozonat Distributed Compression For Still Images

Distributed Compression For Still Images 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

Distributed Compression For Still Images 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

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

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

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

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

Distributed Compression For Still Images 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

Distributed Compression For Still Images 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

Distributed Compression For Still Images 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

Distributed Compression For Still Images 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

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

Distributed Compression For Still Images 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

Distributed Compression For Still Images 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

Distributed Compression For Still Images 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

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

Distributed Compression For Still Images 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

Distributed Compression For Still Images 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

Distributed Compression For Still Images 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

Distributed Compression For Still Images 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