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Transformation

Transformation. Last update on June 15, 2010 Doug Young Suh suh@khu.ac.kr. Entropy and compression. amount of information = degree of surprise Entropy and average code length Information source and coding Memoryless source : no correlation. ∙∙∙∙∙. ∙∙∙.

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Transformation

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  1. Transformation Last update on June 15, 2010 Doug Young Suh suh@khu.ac.kr

  2. Entropy and compression • amount of information = degree of surprise • Entropy and average code length • Information source and coding • Memoryless source : no correlation ∙∙∙∙∙ ∙∙∙ Red blue yellow yellow red black red ∙∙∙ 00011010001100 Media Lab. Kyung Hee University

  3. Entropy • Entropy • What if {0.99,0.003,0.003,0.004}? H(X)≈0 • Reduce H(X)!! • We need narrower pdf. 1/4 1 2 3 4 1/2 1/4 1/8 1 2 3 4 Media Lab. Kyung Hee University

  4. How to getsmall H(X)? • Transformation • Signals in the frequency domain • or mapping into more probable vectors • Predict to reduce uncertainty • H(X) = “degree of uncertainty” • Prediction by using known information Media Lab. Kyung Hee University

  5. Setof ortho-normal vectors • Inner product of two orthogonal vectors is 0. • Inner product • Add multiplications at the same positions. Normal Normal Media Lab. Kyung Hee University

  6. Mapping to { , } • Any vector in the 2 dimensional space can be represented by weighted sumof 2 ortho-normal vectors. • In this case, c0 is stronger. • What does it mean “a weight is large”? weights Media Lab. Kyung Hee University

  7. Howto get the weights? • Inner product for weight • More generally, Media Lab. Kyung Hee University

  8. 2 point DCT/IDCT • 2 point DCT/IDCT • DCT • IDCT Media Lab. Kyung Hee University

  9. 4-pt transform • 4 basis vectors (i.e. code) possible C1 =[½ ½ ½ ½ ] C2 =[½ ½ -½ -½ ] C3 =[½ -½ -½ ½ ] C4 =[½ -½ ½ -½ ] • Matrix representation Networked Video

  10. 8-pt DCT with 8D vectors • Any set of 8 data can be represented by weighted sum of 8 ortho-normal vectors • 8 weights for 8 ortho-normal vectors in the 8 dimensional spaces. • Frequency 0  DC, constant (not varying) Media Lab. Kyung Hee University

  11. 8X8D ortho-normal vectors is calculated for u=1 and x=0~7 (cosine 1/2 period) is calculated for u=2 and x=0~7 (cosine 1 period) is calculated for u=3 and x=0~7 (cosine 1.5 period) is calculated for u=7 and x=0~7 (cosine 3.5 period) Media Lab. Kyung Hee University

  12. 8 8D ortho-normal vectors DC frequency 0 u=0 From low frequency u=1 u=2 u=3 To high frequency n=0  n=3  n=7 Media Lab. Kyung Hee University

  13. 8 8D ortho-normal vectors From low frequency u=4 u=5 u=6 u=7 To high frequency n=0  n=3  n=7 Media Lab. Kyung Hee University

  14. 8-pt DCT/IDCT • IDCT • DCT Media Lab. Kyung Hee University

  15. 8-pt DCT of DC • DC : constant signal 255 255 255 255 255 255 255 255 722 0 0 0 0 0 0 0 Media Lab. Kyung Hee University

  16. 8-pt DCT of stepsignal • Step signal 255 255 255 255 0 0 0 0 360 326 0 -114 0 77 0 65 Media Lab. Kyung Hee University

  17. 8-pt DCT of stepsignal • High frequency signal 255 0 255 0 255 0 255 0 360 65 0 77 0 114 0 327 Media Lab. Kyung Hee University

  18. 8x8 DCT • 2D DCT of the 8x8 block, 64 pixels • 8 point DCT for 8 rows of 8 pixels  8 point DCT for 8 columns • 8x8 DCT • 8x8 IDCT Media Lab. Kyung Hee University

  19. 8x8 DCT • 8 pt DCT is to calculate weights for81Dbasic patterns, • while 8x8 2D DCT is to calculate weights for642Dbasic patterns. • For example,

  20. 8x8 DCT • 64 8x8 patterns U=0 U=7 v=0 v=7

  21. 2D 8x8 DCT • By using 8X8 DCT, 64weights are calculated and stored, respectively. • These are 2D 8x8 DCT coefficients. Media Lab. Kyung Hee University

  22. 2D 8x8 DCT at horizontal edge • F00andF01are large. • For u>0, they are almost 0. DCT Block with Horizontal edge DCT of Horizontal edge Media Lab. Kyung Hee University

  23. 2D 8x8 DCT at verticaledge • F00andF10are large. • For v>0, the values aresmall. • F10< 0 ? DCT Block with vertical edge DCT of vertical edge Media Lab. Kyung Hee University

  24. Low pass filtering (LPF) • Remove weights of higher frequency LPF 0 Media Lab. Kyung Hee University

  25. Energy compaction • More compression when energy distribution is focused to a direction. • The simpler an image, the more compression. • DCT is better thanDFT for image.  KL transform! • Ortho-normal vector patterns of DCT are better suited to image. Simple image Complex image DCT DFT 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 Media Lab. Kyung Hee University

  26. Matrix representation • Matrix representation of 4x4-pt transform where , then • Complexity ~ O(N3)  needs fast algorithm Networked Video

  27. Summary • Transform for compression • Energy compaction • transform = inner product to ortho-normal vectors • weighted sum • Weights = frequency coefficients • No information loss at all !! Media Lab. Kyung Hee University

  28. Video encoding 2-1 2-3 2-2 Original Video + DCT Q VLC Encoded Bitstream IQ Motion vector Motion Estimation Frame Memory IDCT Networked Video

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