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Information theory Data compression perspective

Pasi Fränti. Information theory Data compression perspective. 4.2.2016. Bits and Codes. One bit: 0 and 1 Two bits: 00, 01, 10, 11 Four bits: 0000, 0001, 0010 … 1111 (8 values) Eight bits: 2 256 values (e.g. ASCII code). k bits  2 k values  N values  log 2 N bits. Entropy.

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Information theory Data compression perspective

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  1. Pasi Fränti Information theoryData compression perspective 4.2.2016

  2. Bits and Codes One bit: 0 and 1 Two bits: 00, 01, 10, 11 Four bits: 0000, 0001, 0010 … 1111 (8 values) Eight bits: 2256 values (e.g. ASCII code) k bits  2k values  N values  log2N bits

  3. Entropy Self-entropy of symbol Entropy of source

  4. Prefix code Example of a prefix code a = 0 b = 10 c = 110 d = 111 Example of non-prefix code a = 0 b = 01 c = 011 d = 111 4

  5. Probability distribution 5

  6. Entropy of binary model 6

  7. Huffman coding Codetree Symbols and frequencies First step of the process 7

  8. Huffman coding 8

  9. Two coding methods • Huffman coding • David Huffman, 1952 • Prefix code • Bottom-up algorithm for construction of the code tree • Optimal when probabilities are of the form 2n • Arithmetic coding • Rissanen, 1976 • General: applies to any source • Suitable for dynamic models (no explicit code table) • Optimal for any probability model • All input file is coded as one code word 9

  10. Work space 10

  11. Modeling

  12. Static or adaptive model Static: + No side information + One pass over the data is enough - Fails if the model is incorrect Semi-adaptive: + Optimizes model to the input data - Two-passes over the image needed - Model must also be stored in the file Adaptive: + Optimizes model to the input data + One pass over the data is enough - Must have time to adapt to the data 12

  13. Using wrong model ESTIMATED MODEL: CORRECT MODEL: AVERAGE CODE LENGTH: INEFFICIENCY: 13

  14. Context modelpixel above 14

  15. Context modelpixel to left 15

  16. Summary of context model NO CONTEXT: fw = 56, fb = 8, pw = 87.5 %, pb = 12.5 % Total bit rate = 10.79 + 24 = 34.79 Entropy = 34.79 / 64 = 0.54 bpp PIXEL ABOVE: Total bit rate = 33.28 Entropy = 33.28 / 64 = 0.52 bpp PIXEL TO LEFT: Total bit rate = 7.32 Entropy = 7.32 / 64 = 0.11 bpp 16

  17. Using wrong model

  18. Dynamic modelingState automaton in QM-coder

  19. Example contextsScanned binary images

  20. Effect of context sizeScanned binary images

  21. Arithmetic coding

  22. Block coding • Two problems: • Impossible to make code table for binary input • Cannot use fractions of bits (p=0.9  H=0.07 bits) • Solution 1: Block coding • Block symbols • Contradicts context model • Alphabet explode exponentially with the number of symbols: • 3-symbol blocks 2563=16 M • Solution 2: Arithmetic coding • Block entire input! • No explicit code table

  23. Interval [0,1] dividedup to 3-bits accuracy

  24. Arithmetic coding principles • Length of interval = A • Coding of A takes –log2A bits • Divides the interval according to the probabilities • The lengths of the subintervals sums up to 1. 1 1 c 0.9 b 0.75 0.7 a 0.50 Probabilities: p(a) = 0.7 p(b) = 0.2 p(c) = 0.1 0.25 0 0 25

  25. Coding examplesequence aab Probabilities: p(a) = 0.7 p(b) = 0.2 p(c) = 0.1 A = 0.098 H = -log 0.098 = 3.35 bits 26

  26. Coding of sequence aab Probabilities: 1 1 1 p(a) = 0.7 p(b) = 0.2 p(c) = 0.1 c 0.9 b 0.75 0.75 0.7 0.70 c a 0.63 b 0.49 0.50 0.50 0.490 a c 0.441 b b 0.343 a 0.25 0.25 0 0 0 0 0 27

  27. Code length Length of the final interval: It’s code length: Length with respect to the distribution: 28

  28. Optimality of Arithmetic Coding • Interval is not exactly power of 2. • Round it down to A’ < A that is power of 2 Lower bound for interval size: Upper bound for code length: Length with respect to the distribution: 29

  29. /* Initialize lower and upper bounds */ low 0; high  1; cum[0]  0; cum[1] p1; /* Calculate cumulative frequencies */ FOR i  2 TO k DO cum[i]  cum[i-1] + pk WHILE Symbols left> DO /* Select the interval for symbol c */ c READ(Input); range  high - low; high  low + range*cum[c+1]; low  low + range*cum[c]; /* Initialize lower and upper bounds */ low 0; high  1; cum[0]  0; cum[1] p1; /* Calculate cumulative frequencies */ FOR i  2 TO k DO cum[i]  cum[i-1] + pk WHILE Symbols left> DO /* Select the interval for symbol c */ c READ(Input); range  high - low; high  low + range*cum[c+1]; low  low + range*cum[c]; /* Half-point zooming: lower */ WHILE high < 0.5 DO high  2*high; low  2*low; WRITE(0); FOR buffer TIMES DO WRITE(1); buffer  0; /* Half-point zooming: higher */ WHILE low > 0.5 DO high  2*(high-0.5); low  2*(low-0.5); WRITE(1); FOR buffer TIMES DO WRITE(0); buffer  0; /* Quarter-point zooming */ WHILE (low > 0.25) AND (high < 0.75) THEN high  2*(high-0.25); low  2*(low-0.25); buffer  buffer + 1;

  30. Working space Text box 0.75

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