Computer Vision – Compression(1)

1. Computer Vision – Compression(1) Hanyang University Jong-Il Park

2. Image Compression • The problem of reducing the amount of data required to represent a digital image • Underlying basis • Removal of redundant data • Mathematical viewpoint • Transforming a 2-D pixel array into a statistically uncorrelated data set

3. Topics to be covered • Fundamentals • Basic concepts of source coding theorem • Practical techniques • Lossless coding • Lossy coding • Optimum quantization • Predictive coding • Transform coding • Standards • JPEG • MPEG • Recent issues

4. History of image compression • Theoretic foundation • C.E.Shannon’s works in 1940s • Analog compression • Aiming at reducing video transmission bandwidth  Bandwidth compression • Eg. Subsampling methods, subcarrier modulation… • Digital compression • Owing to the development of ICs and computers • Early 70s: Facsimile transmission – 2D binary image coding • Academic research in 70s to 80s • Rapidly matured around 1990.  standardization such as JPEG, MPEG, H.263, …

5. Data redundancy • Data vs. information • Data redundancy • Relative data redundancy • Three basic redundancies • Coding redundancy • Interpixel redundancy • Psychovisual redundancy

6. Coding redundancy • Code: a system of symbols used to represent a body of information or set of events • Code word: a sequence of code symbols • Code length: the number of symbols in each code word • Average number of bits

7. Eg. Coding redundancy • Reduction by variable length coding

8. Correlation • Cross correlation • Autocorrelation

9. Eg. Correlation

10. Interpixel redundancy • Spatial redundancy • Geometric redundancy • Interframe redundancy

11. Eg. Interpixel redundancy

12. Eg. Run-length coding

13. Psychovisual redundancy +

14. Image compression models • Communication model • Source encoder and decoder

15. Basic concepts in information theory • Self-information: I(E)= - log P(E) • Source alphabet A and symbols • Probability of the events z • Ensemble (A, z) • Entropy(=uncertainty): • Channel alphabet B • Channel matrix Q

16. Mutual information and capacity • Equivocation: • Mutual information: • Channel capacity C • Minimum possible I(z,v)=0 • Maximum possible I over all possible choices of source probabilities in z is the channel capacity

17. 1-pe pbs 0 0 pe pe 1-pbs 1 1 1-pe Eg. Binary Symmetric Channel BSC Entropy Channel capacity Mutual information

18. Noiseless coding theorem • Shannon’s first theorem for a zero-memory source • It is possible to make L’avg/n arbitrarily close to H(z) by coding infinitely long extensions of the source • Efficiency = entropy/ L’avg • Eg. Extension coding • Extension coding  better efficiency

19. Extension coding A B A Efficiency =0.918/1.0=0.918 Better efficiency B Efficiency =0.918*2/1.89=0.97

20. Noisy coding theorem • Shannon’s second theorem for a zero-memory channel: For any R<C, there exists an integer r and code of block length r and rate R such that the probability of a block decoding error is arbitrarily small. • Rate-Distortion theory The source output can be recovered at the decoder with an arbitrarily small probability of error provided that the channel has capacity C > R(D)+e. x feasible x x x Never Feasible!

21. Using mappings to reduce entropy • 1st order estimate of entropy > 2nd order estimate of entropy > 3rd order estimate of entropy …. • The (estimated) entropy of a properly mapped image (eg. “difference source”) is in most cases smaller than that of original image source. How to implement ? The topic of the next lecture!