optimal quantisation

8/11/2019 Optimal Quantisation

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Intensity Level Resolution

Intensity level resolution refers to the number ofintensity levels used to represent the image

The more intensity levels used, the finer the level of

detail discernable in an imageIntensity level resolution is usually given in terms ofthe number of bits used to store each intensity level

Number of Bits Number of IntensityLevels Examples

1 2 0, 1

2 4 00, 01, 10, 11

4 16 0000, 0101, 1111

8 256 00110011, 01010101

16 65,536 1010101010101010


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Intensity Level Resolution (cont)

128 grey levels(7 bpp)



256 greylevels (8 bitsper pixel)

I m a g e s

t a k e n

f r o m

G o n z a l e z

& W o o

d s , D

i g i t a l I m a g e

P r o c e s s

i n g

( 2 0 0 2 )


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16 grey levels (4 bpp)8 grey levels (3 bpp) 4 grey levels (2 bpp) 2 grey levels (1 bpp)


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Image quantization

The quantizers transfer function

Quantizer

t1

Quantizersoutputu u

tk tL+1 t2

r L

r k

r 2

r 1

Quantizationerror

tk+1


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Quantizer

Denote the input brightness range:

Let B the number of bits of the quantizer

L=2 B reconstruction levels

L L L

q

Lt t

q

min Max

L 11

Max L Lt Lt 1min1 ;

maxmin ; L Lu

minmax L L Is the dynamic range of the image


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The optimal (MSE) quantizer (the Lloyd-Max quantizer)

Lk duu pr ur

t pr t r t t

k

k

t

t uk

k

k uk k k k k

10)()(2

0)()()(

1

221

L

k

t

t uk

M

m

N

n

k

k

duu pr unmunmu MN 1

21

0

1

0

21

)()(,',1

Leibnitz rule : differentiation under an integral


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t r r

k k k 1

2

k t

t

u

t

t

u

k u|u E

(u)du p

(u)duup

r 1k

k

1k

k

Iterative solution !


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The uniform quantizer = the optimal quantizer for the uniformgrey level distribution:

otherwise

t ut t t u p L

Lu

0

,1

)( 11

11

r t t t t

t t k

k k

k k

k k

( )

( )1

2 2

1

1

2 2

t t t

k k k 1 1

2

2

qt r k k


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1

122

2

2 2

qu du

q

q

q

/

/

dB B6

210log SNR

22B

10

2B

2u

121 22/

2/

22 Aduu A

A

Au

Signal variance

Noise variance

SNR


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(Laplacian)

variance , - mean

2

2

2 2)(

exp2

1)(

uu pu

uu pu exp2)(

22

(Gaussian)

Other possible distributions


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Illustration of uniform quantization

B=1 => L=2

t1=0 t2=128 t3=256

r1=64

r2=192

Uniform quantizer transfer function

Decision levels

R

e c o n s

t r u c

t i o n

l e v e

l s

0 50 100 150 200 250

0

100

200

300

400

500

600

700

800

900

1000

The histogram of the image


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B=2 => L=4

Quantized imageThe histogram of the image

0 50 100 150 200 250

0

100

200

300

400

500

600

700

800

900

1000


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B=3 => L=8; false contours present

Quantized imageImage histogram

0 50 100 150 200 250

0

100

200

300

400

500

600

700

800

900

1000



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0 50 100 150 200 250

0

100

200

300

400

500

600

700

800

900

1000

Illustration of optimal quantization

Input image histogram

Quantized image (Lloyd max algo)

B=1 => L=2


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0 50 100 150 200 250

0

100

200

300

400

500

600

700

800

900

1000

Illustration of optimal quantizationB=3 => L=8

Input image histogramQuantized image (Lloyd max algo)


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Uniform quantization, B=4Uniform quantization, B=6

False Contouring


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16 grey levels (4 bpp) 8 grey levels (3 bpp)4 grey levels (2 bpp) 2 grey levels (1 bpp)

False Contouring


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Resolution: How Much Is Enough?

The big question with resolution is always howmuch is enough ?

This all depends on what is in the image and

what you would like to do with itKey questions include

Does the image look aesthetically pleasing?Can you see what you need to see within theimage?


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Resolution: How Much Is Enough?

The picture on the right is fine for counting thenumber of cars, but not for reading the numberplate


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Intensity Level Resolution(cont)

I m a g e s

t a k e n

f r o m

G o n z a

l e z

& W o o

d s , D

i g i t a l I m a g e

P r o c e s s

i n g

( 2 0 0 2 )

Low Detail Medium Detail High Detail


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)(

)(

)((

)(

))((

))(()'(

)'(

1

1

1

1

1

1

1

1

1

u E

duuup

duuup

duu p

duuup

duu pu E

r pu E

L

k

k

k

k

k

k

k

k

t

t u

L

k

t

t u

t

t u

t

t

u L

k

t

t u

k

L

k k

Quantizer output unbiased

estimator of the input


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))'((

)|()|'()'(

2

2

1

1

u E

r p

uu E uu E puu E

k

L

k k

k k

L

k

k

Quantization error uncorrelatedwith the output (principle oforthogonality)


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)()())(()(

))(()())'((

2

2'22

2'22

'

E u E u E u E

u E u E uu E

uu

Average power (variance) ofquantizer output is less thanthat of input

optimal quantisation

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