University of the PhilippinesAugust 2006Diane LingrandUniversity of Nice – Sophia Antipolis, France Quantification

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2006

Quantification

University of the Philippines August 2006

Diane Lingrand

University of Nice – Sophia Antipolis, France

lingrand@polytech.unice.fr

http://www.polytech.unice.fr/~lingrand

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2006

Quantification

• Only discrete values are available

• Usually:

– grey levels: integers from 0 to 255

– color components: integers from 0 to 255

= 256 different values

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2006

Information coding

• 1 bit values can be 0 or 1 : 2 different values (=2¹)

• 2 bits: 00, 01, 10, 11: 4 different values (=2²)

• 3 bits: 000, 001, 010, 011, 100, 101, 110, 111:

8 different values (=2³)

• ....

• 1 byte = 8 bits: 256 = 2⁸ different values

– to store integers from 0 to 255, we need 1 byte

• ...

• n bits: 2ⁿ different values

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2006

Reducing the number of bits used ...

• ... will reduce the number of colors.

• How to choose the colors ?

255 255

0 f

g

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2006 ⁵

Principle

• decision level : d_i

• reconstruction level : r_j

d₀ d₁ d₂

…

d_n1 d_n

min(f)=m r₀ r₁ r_n1 max(f)=M

d_i d_i+1 r_i

f g

r_i

if f(x,y) ∈ [d_i; d_i+1[ then g(x,y)=r_i

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2006 ⁶

Quantification error

• we want to minimize error between f and g

• values of g : samples of a random process with probability density p(f)

• hypothesis : probability density of each quantification level is constant : p(r

_i

)

H

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2006

Consequences of hypothesis

• Then :

H

and finally :

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2006 ⁸

How to decide the reconstruction levels ?

• we want to minimize error E with respect to r_i – determination of r_i

then :

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2006 ⁹

How to decide the reconstruction levels ?

with

[Panter et Dite]

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2006 ¹⁰

Uniform probability density :

• constant length quantification intervals

• centered quantification levels

0 255

255

k*

256/2ⁿ

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2006 ¹¹

General case

• Hypothesis is not verified

• Recursive formulae

• Tables for uniform probability densities, gaussian, ...

H

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2006 ¹²

Grey levels quantification

• number of quantification levels: 2ⁿ where n is the number of bits

• 8 bits : 256 grey levels

• human visual capacity : 10 to 15 absolute grey levels. Many more in relative.

2 bits

1 bit 3 bits

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2006 ¹³

Colors quantification

• considering the components separately – number of bits may differ

• RGB seems to be the more appropriate

• ex: SVGA :

– (R, G, B) = (5 bits, 5 bits, 5 bits)

• on your computer ?

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2006 ¹⁴

8 bits , 256 grey levels

16 777 216 colors

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2006 ¹⁵

7 bits : 128 grey levels

2 097 142 colors

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2006 ¹⁶

262 144 colors

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2006 ¹⁷

37 768 colors

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2006 ¹⁸

4 096 colors

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2006 ¹⁹

512 colors

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2006 ²⁰

64 colors

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2006 ²¹

1 bit : 2 grey levels

8 colors

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2006 ²²

4 bits. logarithmic quantification

example :

255 * log(x/10+1)/log(26.5)

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2006

Bibliography

•“Digital Image Processing" par W. K. Pratt, John Wiley & Sons, inc., Third Edition, 2001

•"Digital Image Processing" par Gonzalez et Woods, Prentice Hall, Second Edition, 2002

•http://www.dai.ed.ac.uk/HIPR2/

•Books available on the web

http://homepages.inf.ed.ac.uk/rbf/CVonline/books.htm

•Computer Vision Online

http://www.dai.ed.ac.uk/CVonline/transf.htm

•Vetterli's talks (wavelets, ..)

http://lcavwww.epfl.ch/~vetterli/talks/index.html