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

Quantization and error

Last updated on June 15, 2010

Doug Young Suh

[email protected]


Entropy and compression

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

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

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


Quantization1

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

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 is 6 02db

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

Effect of quantization in image

DCT

Q

Q-1

IDCT

IDCT

PSNRInf

PSNR25dB

Media Lab. Kyung Hee University


Pdf and quantization error

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

Non-uniform pdf

  • Variable step size

     Less error

  • Fixed step size

     More error

Media signal


Error for fixed step size

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

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

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

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

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 dpcm1

Differential Pulse-Coded Modulation (DPCM)

One - Tap Prediction

N – Tap Prediction

Quantize

Prediction

Media signal


Quantization and error

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

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

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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