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Quantization and error

Quantization and error. Last updated 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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Quantization and error

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  1. Quantization and error Last updated 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. Fine-to-coarse Quantization • Dice vs. coin • Effects of quantization • Data compression • Information loss, but not all 1/6 1/2 {1,2,3}  head {4,5,6}  tail H T 1 2 3 4 5 6 quantization 3 5 2 1 5 4 ∙∙∙ H T H H T T ∙∙∙ Media Lab. Kyung Hee University

  4. Quantization • analog-to-digit-al quantization • In order to cook in binary computers • digital TV, digital comm., digital control… • fine-to-coarse digital quantization ADC Infinite numbers finite numbers Media Lab. Kyung Hee University

  5. Quantization • Digital • Selectable accuracy : scale for human vs. gold • [dynamic range, required accuracy, pdf] • open questions • Weights of soldiers are ranged from50 kg to100 kg, while that of new born baby is less than 5kg. • Voice signal of mobile phones is quantized in 8bits, while CD quality audio is quantized in 16bits. Why is 8bits enough for voice? Media Lab. Kyung Hee University

  6. Quantization/de-quantization • Representing values and error(-5kg ~ 5kg) • x1=50.341kg, x2=67.271kg, x3=45.503kg, x4=27.91kg, …  000 010 001 111 Media Lab. Kyung Hee University

  7. Effect of 1 additional is6.02dB • Dynamic range of R, B bits • Step size Δ = R/2B • Quantization noise power = E[e2] • Noise in dB (log102=3.01) probability 1/Δ e -Δ/2 Δ/2 Media Lab. Kyung Hee University

  8. Effect of quantization in image DCT Q Q-1 IDCT IDCT PSNRInf PSNR25dB Media Lab. Kyung Hee University

  9. pdf and quantization error • pdf (probability density function) • The narrower pdf, the less number of bits at the same error • The narrower pdf, the less error at the same number of bits Media signal

  10. Non-uniform pdf • Variable step size  Less error • Fixed step size  More error Media signal

  11. Error for fixed step size • Representing values at all intervalsare -0.75, -0.25, 0.25, 0.75, respectively, then mean square errors become, Media signal

  12. Error for variable step size • What representing value minimizes mean square error in each interval? • For example, in the interval 00, the following equation is differentiated by p to find minimum. Media signal

  13. Correlation in text • memory-less and memory I(x) = log2 (1/px) = “degree of surprise” • qu-, re-, th-, -tion, less uncertain • Of course, there are exceptions... Qatar, Qantas • Conditional probability • p(u|q) >> p(u) • Then, I(u|q) << I(u) • accordingly, I(n|tio) << I(n) Media Lab. Kyung Hee University

  14. Differential Pulse-Coded Modulation (DPCM) • Quantize not x[n] but d[n]. • Principle: Pdf of d[n] is narrower than that of x[n]. • Less error at the same number of bits. • Less amount of data, at the same error. Quantize Prediction Media signal

  15. Effects of DPCM • Histograms in images simple imagecomplex image Prob. Prob. x[n] x[n] Q Prob. Prob. H(D1)<H(D2) Pred 0 0 d[n] d[n] Media Lab. Kyung Hee University

  16. Differential Pulse-Coded Modulation (DPCM) One - Tap Prediction N – Tap Prediction Quantize Prediction Media signal

  17. DPCM • Determine “a” which minimizes where R(1) is the auto-correlation for zero mean signal a << 0 a > 0 a ≈ 0 time Media Lab. Kyung Hee University

  18. Adaptive DPCM • Prediction filter coefficients are estimated periodically and sent as side information. • CDMA IS-95, CELP, EVRC (update interval 50 or 100 ms) LPC (linear predictive coding) • Drawbacks 1. Correlation should be given and stationary. 2. Error propagation : needs refreshment • Open questions • Why is quantized difference used for prediction? • Will quantization noise be accumulated? • How often do we have to refresh? • How about non-stationary case? Media signal

  19. Summary • Trade-off between bit-rate and quality • [dynamic range, accuracy, pdf] • Narrower pdf is preferred, w.r.t. H(X) • Prediction for narrower pdf • Widely used in audio-video codecs • Adaptation for better prediction • Error propagation Media Lab. Kyung Hee University

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