Quantization and error
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Quantization and error. Last updated on June 15, 2010 Doug Young Suh [email protected] 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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Quantization and error

Last updated on June 15, 2010

Doug Young Suh

[email protected]


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


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


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


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


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


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


Effect of quantization in image

DCT

Q

Q-1

IDCT

IDCT

PSNRInf

PSNR25dB

Media Lab. Kyung Hee University


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


Non-uniform pdf

  • Variable step size

     Less error

  • Fixed step size

     More error

Media signal


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


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


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


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


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


Differential Pulse-Coded Modulation (DPCM)

One - Tap Prediction

N – Tap Prediction

Quantize

Prediction

Media signal


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


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


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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