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4C8 Dr. David Corrigan

4C8 Dr. David Corrigan. Jpeg and the DCT. 2D DCT. DCT Basis Functions . 2D DCT. Q step = 15. Each band is the same size and there are 64 bands in total so the entropy is. Optimum Block Size is 8!. Slow DCT. Sledgehammer implementation for 8 point DCT

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4C8 Dr. David Corrigan

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  1. 4C8Dr. David Corrigan Jpeg and the DCT

  2. 2D DCT DCT Basis Functions

  3. 2D DCT

  4. Qstep = 15

  5. Each band is the same size and there are 64 bands in total so the entropy is

  6. Optimum Block Size is 8!

  7. Slow DCT • Sledgehammer implementation for 8 point DCT • Each row multiply requires 8 MADDs (approx) • So for all 8 rows requires 64 MADDs (approx)

  8. Fast DCT • Exploit Symmetry

  9. Fast DCT • So split Matrix T into two parts...

  10. Fast DCT • split Matrix T into two parts, change y...

  11. Fast DCT 4 “adds”, 16 MADDS for each operation = 8 adds and 32 MADDS = 40 ops Compare with 64 MADDS from before .

  12. Fast DCT This sub-matrix can be simplified with symmetry again! 4 “adds”, 8 MADDS in total = 12 ops (down from 20) So now we are at 20 (for the first sub matrix) + 12 (for these two) = 32 ops So we have saved about x2!

  13. JPEG and Colour Images • JPEG uses YCBCRcolourspace. • The chrominance channels are usually downsampled. • There are 3 commonly used modes • 4:4:4 – no chrominance subsampling • 4:2:2 – Every 2nd column in the chrominance channels are dropped. • 4:2:0 – Every 2nd column and row is dropped.

  14. Subjectively Weighted Quantisation • In JPEG it is standard to apply different thresholds to different bands

  15. Subjectively Weighted Quantisation • These values are obtained by perceptual tests. • A user is asked to view an image of a particular size on at specified distance from the screen. • Usually a multiple of the screen height. • User is presented with an image and is asked to increase the gain of a given band until he/she just notices a difference in the image. • Note typically a flat grey image is used to avoid masking effects caused by edges and texture • The set of form the quantisation matrix.

  16. Subjectively Weighted Quantisation • Lower Frequency Bands are assigned lower step sizes. • There is a slight drop of in step size from the DC coefficient to low frequency coefficients. • The step sizes for the chrominance channels increase faster than for luminance.

  17. We have seen this before

  18. Comparing Different Quantisations JPEG Uncompressed Qstep = Qlum

  19. Comparing Different Quantisations Qstep = Qlum PSNR = 32.9 dB

  20. Comparing Different Quantisations JPEG Uncompressed Qstep = 2 * Qlum PSNR = 30.6 dB

  21. Comparing Different Quantisations Qstep = 15 Qstep = Qlum Qstep = 15 PSNR = 37.6 dB

  22. Comparing Different Quantisations Qstep = 30 Qstep = Qlum Qstep = 30 PSNR = 33.4 dB

  23. Comparing Different Quantisations PSNR indicates better quality for Qstep = 30 over Qstep = Qlum but this clearly is not true from a subjective analysis. Qstep = 30 Qstep = Qlum Qstep = 30 PSNR = 33.4 dB

  24. Comparing Different Quantisations Using the subjectively weighted Quantisation achieves much higher levels of compression for equivalents levels of quality.

  25. JPEG Coding • The most obvious way might seem to code each band separately • ie. Huffman with RLC like we suggested with the Haar Transform. • We could get close to the entropy • This is not the way it is coded because • It would require 64 different codes. High cost in computation and storage of codebooks. • It ignores the fact that the zero coefficients occur at the same positions in multiple bands.

  26. JPEG Coding • Instead we code each block separately • A block contains 64 coefficients, one from each band. • Each block contains 1 DC coefficient (from the top left band) and 63 AC coefficients • Two codebooks are used in total for all the blocks, one for the DC coefficients and the other for the AC coefficients. • At the end of each Block we insert an End Of Block (EOB) symbol in the datastream

  27. Data Ordering • Each block covers is a 8x8 grid of coeffs • A Zig-Zag scan converts them into a 1D stream. • As most non-zero values occur in the top left corner using a Zig-Zag scan maximises the lengths zero runs so improves efficiency of RLC

  28. Zig-Zag Scan Example Non-Zero values are at the top left corner of the block Zig-Zag scan concentrates the non-zero coefficients at the start of the stream -13, -3, 6, 0, 0, 2, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 36 more zeros, the end Typical DCT Block Coefficients

  29. Coding the DC Coefficients Differential Coding

  30. Coding the DC Coefficients This value is actually the difference between the dc coefficient of the current and previous blocks Typical DCT Block Coefficients

  31. Coding DC Coefficients • There is potentially a large number of levels to encode. • Up to 4096 depending on the quantization step size. • We break down the symbol value into a size index pair

  32. Coding DC Coefficients • So if the DC value is -13 • The size is 4 • The index is 0010 • In JPEG only the size is encoded using Huffman • The index is uncoded, efficiency is not dramatically affected. • Only 12 codes required in huffman table • Table size is 16 + 12 = 28 bytes

  33. More examples of Coefficient to size/index pair conversions

  34. Coding the AC Coefficients Size/Index Pair for DC coefficient The length of the run and the value of the coeff after it are strongly correlated 40010, -3, 6, 2 zeros, 2, 3 zeros, -1, 17 zeros, 1, 36 more zeros, the end The block usually ends with a long run of zeros Typical DCT Block Coefficients

  35. Coding the AC Coefficients • Code/Size Correlations • High coeffs follow short runs and low coeffs follow long runs • Final run of zeros • These don’t need to be coded • Just tell the encoder that there are no more non-zero coefficients and move onto the next block.

  36. Symbols Run/Coefficient Symbols eg. 0, 0, 9 is a run of 2 zeros followed by a 9 However we represent 9 using the size/index format from the dc coeffs 9 has a size of 4 and an index 1001 So we code the run/size pair (2,4) and the index 1001 is appended to the stream

  37. Symbols • Run/Size Symbols • All possible combinations of runs from 0->15 and size from 1->10 • 160 total symbols • Huffman Codes are used for each symbol • Index values are not coded further

  38. Special Symbols • ZRL • Used to represent a run of 16 zeros • Used when the run of zeros is greater than 15 • Eg. 17 zeros, 14 - is coded as (ZRL) (1,4) 1110 • EOB • Inserted when a block ends with a run of zeros In total there are 160 run/size symbols and 2 special symbols 162 symbols to 2 encode codetable is 16 + 162 = 178 bytes

  39. Coding Example -13, -3, 6, 2 zeros, 2, 3 zeros, -1, 17 zeros, 1, 36 more zeros, the end DC Coefficient is -13. The size is 4 and the index is 0010 Typical DCT Block Coefficients Current Stream State: 40010

  40. Coding Example -13, -3, 6, 2 zeros, 2, 3 zeros, -1, 17 zeros, 1, 36 more zeros, the end The first ac value is -3. That is a run of 0 zeros followed by -3. -3 has size 2 and index 0000 Therefore the run/size pair is (0,2) Current Stream State: 40010 (0,2) 00

  41. Coding Example -13, -3, 6, 2 zeros, 2, 3 zeros, -1, 17 zeros, 1, 36 more zeros, the end The next ac value is 6. That is a run of 0 zeros followed by 6. 6 has size 3 and index 110 Therefore the run/size pair is (0,3) Current Stream State: 40010 (0,2) 00 (0,3) 110

  42. Coding Example -13, -3, 6, 2 zeros, 2, 3 zeros, -1, 17 zeros, 1, 36 more zeros, the end The next ac value to encode is a run of 2 zeros followed by a ac coefficient 2. 2 has size 2 and index 10 Therefore the run/size pair is (2,2) Current Stream State: 40010 (0,2) 00 (0,3) 110 (2,2) 10

  43. Coding Example -13, -3, 6, 2 zeros, 2, 3 zeros, -1, 17 zeros, 1, 36 more zeros, the end The next ac value to encode is a run of 3 zeros followed by a ac coefficient -1. -1 has size 1 and index 0 Therefore the run/size pair is (3,1) Current Stream State: 40010 (0,2) 00 (0,3) 110 (2,2) 10 (3,1)0

  44. Coding Example -13, -3, 6, 2 zeros, 2, 3 zeros, -1, 17 zeros, 1, 36 more zeros, the end The next ac value to encode is a run of 17 zeros followed by a ac coefficient 1. As the run is > 15 zeros we have to use the ZRL symbol to code the first 16 zeros. The remaining run length consists of (17 - 16) = 1 zero. An ac coefficient of 1 has size 1 and index 1 Therefore we insert the run/size pair (1,1) after the ZRL marker Current Stream State: 40010 (0,2) 00 (0,3) 110 (2,2) 10 (3,1)0 ZRL (1,1) 1

  45. Coding Example -13, -3, 6, 2 zeros, 2, 3 zeros, -1, 17 zeros, 1, 36 more zeros, the end The remaining coeffs are all 0. Therefore the EOB marker is used. If the last ac coeff is non-zero, then the EOB marker is not used. Current Stream State: 40010 (0,2) 00 (0,3) 110 (2,2) 10 (3,1)0 ZRL (1,1) 1 EOB

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