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Binary Image Compression

Binary Image Compression. The art of modeling image source Image pixels are NOT independent events Run length coding of binary and graphic images Applications: BMP, TIF/TIFF Lempel-Ziv coding* How does the idea of building a dictionary can achieve data compression?. From Theory to Practice.

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Binary Image Compression

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  1. Binary Image Compression • The art of modeling image source • Image pixels are NOT independent events • Run length coding of binary and graphic images • Applications: BMP, TIF/TIFF • Lempel-Ziv coding* • How does the idea of building a dictionary can achieve data compression? EE465: Introduction to Digital Image Processing

  2. From Theory to Practice • So far, we have discussed the problem of data compression under the assumption that the source distribution is known (i.e., the probabilities of all possible events are given). • In practice, the source distribution (probability models) is unknown and can only be approximately obtained by relative frequency counting method due to the inherent dependency among source data. EE465: Introduction to Digital Image Processing

  3. An Image Example Binary image sized 100100 (approximately 1000 dark pixels) EE465: Introduction to Digital Image Processing

  4. A Bad Model • Each pixel in the image is observed as an independent event • All pixels can be characterized by a single discrete random variable – binary Bernoulli source • We can estimate the probabilities by relative frequency counting EE465: Introduction to Digital Image Processing

  5. Synthesized Image by the Bad Model Binary Bernoulli distribution (P(X=black)=0.1) EE465: Introduction to Digital Image Processing

  6. Why does It Fail? • Roughly speaking • Pixels in an image are not independent events (source is not memoryless but spatially correlated) • Pixels in an image do not observe the same probability model (source is not stationary but spatially varying) • Fundamentally speaking • Pixels are the projection of meaningful objects in the real world (e.g., characters, lady, flowers, cameraman, etc.) • Our sensory processing system has learned to understand meaningful patterns in sensory input through evolution EE465: Introduction to Digital Image Processing

  7. Similar Examples in 1D • Scenario I: think of a paragraph of English texts. Each alphabet is NOT independent due to semantic structures. For example, the probability of seeing a “u” is typically small; however, if a “q” appears, the probability of the next alphabet being “u” is large. • Scenario II: think of a piece of human speech. It consists of silent and voiced segments. The silent segment can be modeled by white Gaussian noise; while the voiced segment can not (e.g., pitches) EE465: Introduction to Digital Image Processing

  8. Data Compression in Practice Y entropy coding binary bit stream source modeling discrete source X P(Y) probability estimation Probabilities can be estimated by counting relative frequencies either online or offline The art of data compression is the art of source modeling EE465: Introduction to Digital Image Processing

  9. Source Modeling Techniques • Transformation • transform the source into an equivalent yet more convenient representation • Prediction • Predict the future based on the causal past • Pattern matching • Identify and represent repeated patterns EE465: Introduction to Digital Image Processing

  10. Non-image Examples • Transformation in audio coding (MP3) • Audio samples are transformed into frequency domain • Prediction in speech coding (CELP) • Human vocal tract can be approximated by an auto-regressive model • Pattern matching in text compression • Search repeated patterns in texts EE465: Introduction to Digital Image Processing

  11. How to Build a Good Model • Study the source characteristics • The origin of data: computer-generated, recorded, scanned … • It is about linguistics, physics, material science and so on … • Choose or invent the appropriate tool (modeling technique) Q: why pattern matching is suitable for texts, but not for speech or audio? EE465: Introduction to Digital Image Processing

  12. Image Modeling • Binary images • Scanned documents (e.g., FAX) or computer-generated (e.g., line drawing in WORD) • Graphic images • Windows icons, web banners, cartoons • Photographic images • Acquired by digital cameras • Others: fingerprints, CT images, astronomical images … EE465: Introduction to Digital Image Processing

  13. Lossless Image Compression • No information loss – i.e., the decoded image is mathematically identical to the original image • For some sensitive data such as document or medical images, information loss is simply unbearable • For others such as photographic images, we only care about the subjective quality of decoded images (not the fidelity to the original) EE465: Introduction to Digital Image Processing

  14. Binary Image Compression • The art of modeling image source • Image pixels are NOT independent events • Run length coding of binary and graphic images • Applications: BMP, TIF/TIFF • Lempel-Ziv coding* • How does the idea of building a dictionary can achieve data compression? EE465: Introduction to Digital Image Processing

  15. Run Length Coding (RLC) What is run length? Run length is defined as the length of consecutively identical symbols Examples HHHHHTHHHHHHH Coin-flip 5 1 7 SSSSEEEENNNNWWWW random walk 4 4 4 4 EE465: Introduction to Digital Image Processing

  16. Run Length Coding (Con’t) Y Transformation by run-length counting Entropy coding binary bit stream discrete source X P(Y) probability estimation Y is the sequence of run-lengths from which X can be recovered losslessly EE465: Introduction to Digital Image Processing

  17. RLC of 1D Binary Source 00001110000001111100 X 4 3 6 5 2 Y Huffman coding (need extra 1 bit to denote what the starting symbol is) compressed data Properties • “0” run-length (red) and “1” run-length (green) alternates - run-lengths are positive integers EE465: Introduction to Digital Image Processing

  18. Variation of 1D Binary RLC When P(x=0) is close to 1, we can record run-length of dominant symbol (“0”) only Example 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 11 0 0 0 0 0 0 0 0 1 … 5 7 4 0 8 run-length Properties - all coded symbols are “0” run-lengths - run-length is a nonnegative integer EE465: Introduction to Digital Image Processing

  19. Modeling Run-length geometric source: P(X=k)=(1/2)k, k=1,2,… Run-length k Probability 1/2 1/4 1/8 1/16 1/32 … 1 2 3 4 5 … EE465: Introduction to Digital Image Processing

  20. Golomb Codes Optimal VLC for geometric source: P(X=k)=(1/2)k, k=1,2,… codeword 0 10 110 1110 11110 111110 1111110 11111110 … … k 1 2 3 4 5 6 7 8 … 0 1 1 0 1 0 1 0 … EE465: Introduction to Digital Image Processing

  21. From 1D to 2D white run-length black run-length Question: what distinguishes 2D from 1D coding? Answer: inter-line dependency of run-lengths EE465: Introduction to Digital Image Processing

  22. Relative Address Coding (RAC)* previous line 0 0 0 0 01 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 7 current line 0 0 0 0 00 1 1 1 1 1 1 1 10 0 0 0 0 1 1 0 0 d1=1 d2=-2 NS,run=2 NS – New Start Its variation was adopted by CCITT for Fax transmission EE465: Introduction to Digital Image Processing

  23. Image Example CCITT test image No. 1 Size: 17282376 Raw data (1bps) filesize of ccitt1.pbm: 513216 bytes filesize of ccitt1.tif: 37588 bytes Compression Ratio=13.65 EE465: Introduction to Digital Image Processing

  24. Graphic Images (Cartoons) Total 12 different colors (black,white,red,green,blue,yellow …) • Observations: • dominant background color (e.g., white) • objects only contain a few other colors EE465: Introduction to Digital Image Processing

  25. Palette-based Image Representation color index 0 white Any (R,G,B) 24-bit color can be repre- sented by its index in the palette. 1 black 2 red green 3 4 blue yellow 5 … … EE465: Introduction to Digital Image Processing

  26. RLC of 1D M-ary Source Basic idea Record not only run-lengths but also color indexes Example color sequence WWWWGGGWWWWWWWRRRR… 0 0 0 0 03 3 30 0 0 0 0 0 02 2 2 210 0 0 0 0 0 0 0 … 5 3 7 4 1 8 run-length 0 3 0 2 1 0 color-index (color, run) representation: (0,5) (3,3) (0,7) (2,4) (1,1) (0,8) … Note: run-length is a positive integer EE465: Introduction to Digital Image Processing

  27. Variation of 1D M-ary RLC When P(x=0) is close to 1, we can record run-length of dominant symbol (“0”) only Example 0 0 0 0 0 1 0 0 0 0 0 0 0 4 0 0 0 0 32 0 0 0 0 0 0 0 0 1 … 5 7 4 0 8 run 1 4 3 2 1 level (run, level) representation: (5,1) (7,4) (4,3) (0,2) (8,1) … Properties - “0” run-lengths only - run-length is a nonnegative integer EE465: Introduction to Digital Image Processing

  28. Image Example Raw data: party8.ppm, 526286, 451308 bytes Compressed: party8.bmp, 17094 bytes Compression ratio=26.4 EE465: Introduction to Digital Image Processing

  29. Binary Image Compression • The art of modeling image source • Image pixels are NOT independent events • Run length coding of binary and graphic images • Applications: BMP, TIF/TIFF • Lempel-Ziv coding* • How does the idea of building a dictionary can achieve data compression? EE465: Introduction to Digital Image Processing

  30. History of Lempel-Ziv Coding • Invented by Lempel-Ziv in 1977 • Numerous variations and improvements since then • Widely used in different applications • Unix system: compress command • Winzip software (LZW algorithm) • TIF/TIFF image format • Dial-up modem (to speed up the transmission) EE465: Introduction to Digital Image Processing

  31. Dictionary-based Coding • Use a dictionary • Think about the evolution of an English dictionary • It is structured - if any random combination of alphabets formed a word, the dictionary would not exist • It is dynamic - more and more words are put into the dictionary as time moves on • Data compression is similar in the sense that redundancy reveals as patterns, just like English words in a dictionary EE465: Introduction to Digital Image Processing

  32. Toy Example I took a walk in town one day And met a cat along the way. What do you think that cat did say? Meow, Meow, Meow entry pattern 1 I took a walk in town one day 2 And met a I took a walk in town one day And met a pig along the way. What do you think that pig did say? Oink, Oink, Oink 3 along the way 4 What do you think that 5 did say? I took a walk in town one day And met a cow along the way. What do you think that cow did say? Moo, Moo, Moo 6 cat 7 meow … … … - from “Wee Sing for Baby” EE465: Introduction to Digital Image Processing

  33. Basic Ideas • Build up the dictionary on-the-fly (based on causal past such that decoder can duplicate the process) • Achieve the goal of compression by replacing a repeated string by a reference to an earlier occurrence • Unlike VLC (fixed-to-variable), LZ parsing goes the other way (variable-to-fixed) EE465: Introduction to Digital Image Processing

  34. Lempel-Ziv Parsing • Initialization: • D={all single-length symbols} • L is the smallest integer such that all codewords whose length is smaller than L have appeared in D (L=1 at the beginning) • Iterations: wnext parsed block of L symbols • Rule A: If wD, then represent w by its entry in D and update D by adding a new entry with w concatenated with its next input symbol • Rule B: If wD, then represent the first L-1 symbols in w by its entry in D and update D by adding a new entry with w EE465: Introduction to Digital Image Processing

  35. Example of Parsing a Binary Stream Dictionary input: 0 1 1 0 0 1 1 1 0 1 1 … entry pattern 1 0 w: 0, 11, 10, 00, 01, 110, 011, … 2 1 rule: L=1 A B B B A B A 3 01 output: 1, 2, 2, 1, 3, 4, 7, … (entries in D) 11 4 L=2 Illustration: Step 1: w=0, Rule A, output 1, add 01 to D, L=2 Step 2: w=11, Rule B, output 2, add 11 to D Step 3: w=10, Rule B, output 2, add 10 to D Step 4: w=00, Rule B, output 1, add 00 to D Step 5: w=01, Rule A, output 3, add 011 to D, L=3 Step 6: w=110, Rule B, output 4, add 110 to D 10 5 6 00 011 7 L=3 110 8 fixed -length variable -length EE465: Introduction to Digital Image Processing

  36. Binary Image Compression Summary • Theoretical aspect • Shannon’s source entropy formula tells us the lower bound for coding a memoryless discrete source • To achieve the source entropy, we need to use variable length codes (i.e., long codeword assigned to small-probability event and vice versa) • Huffman’s algorithm (generating the optimal prefix codes) offers a systematic solution • Practical aspect • Data in the real world contains memory (dependency, correlation …) • We don’t know the probability distribution function of the source • Tricky point: the goal of image compression (processing) is not to make your algorithm work for one image, but for a class of images EE465: Introduction to Digital Image Processing

  37. The Art of Source Modeling How do I know a model matches or not? Hypothesis Model: Bernoulli distribution (P(X=black)=0.1) Observation Data EE465: Introduction to Digital Image Processing

  38. A Matched Example Synthesis Model EE465: Introduction to Digital Image Processing

  39. Good Models for Binary Images Transformation by Lempel-Ziv parsing Y Transformation by run-length counting Huffman coding binary bit stream discrete source X P(Y) probability estimation Y is the sequence of run-lengths from which X can be recovered losslessly Y is the sequence of dictionary entries from which X can be recovered losslessly EE465: Introduction to Digital Image Processing

  40. Remaining Questions • How to choose different compression techniques, say RLC vs. LZC? • Look at the source statistics • For instance, compare binary image (good for RLC) vs. English texts (good for LZC) • What is the best binary image compression technique? • It is called JBIG2 – an enhanced version of JBIG • What is the entropy of binary image source? • We don’t know and the question itself is questionable • A more relevant question would be: what is the entropy for a probabilistic model which we believe generates the source? EE465: Introduction to Digital Image Processing

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