QUANTIZATION
by
Joseph RONSIN
2
Outlines
Definition
Scalar quantization
Definition
Distortion
Non uniform quantization
Quantization CH 4
3
Discrete
values
Y = Q(x)
Continuous
amplitude
x
Quantization
Definition:
Discretisation of color space
One value for a set of values on an interval
Defining number of intervals
Depends on display (physical factors)
Depends on human visual properties (SVH)
Quantization
4
Topics:
Acquisition
Processing: reduction of number of grey levels or
colors inside original image
Minimal Distorsion
Interest
Reduction of number of bits
Displaying a picture with N bits on a display with M<N
bits
Quantization types
Scalar (linear or not)
Vectorial
Quantization CH 4
5
Type of images
Different Quantizations and corresponding types
of images
binary
I (x,y)
{ 0,1 }
Monochrome or grey
I (x,y)
[ a, b ]
often a = 0 et b = 255
color RVB
I
r(x,y)
I (x,y) =
I
v(x,y)
I
b(x,y)
black
white
black
white
6
Outlines
Definition
Scalar quantization
Definition
Distortion
Non uniform quantization
Quantization CH 4
7
Definition:
Quantization levels
max
iL
max
min
min
0
1
i
i
i
x
d
x
,
x
x
with
x
d
1
L
,...,
0
i
d
x
d
if
q
)
x
(
Q
q
i-1q
iq
i+1d
i-1d
id
i+1Output Qx
Input X
decision thresholds
Scalar Quantization
8
Other representation: Quantization Characteristic
Uniform Quantization – N reconstruction levels
Q(x)
X : Input
q
id
id
i+1Output
q
N/2d
N/2-1Scalar Quantization
Quantization CH 4
9
Distorsion
Measure of distorsion: objective criterion
For an original image MxN represented with B bits
dynamic of symbols :
2
B-1 = L
x
i,jand
y
i,ipixels of original image and quantized image
MSE (Mean Square Error)
PSNR (Peak Signal-to-Noise Ratio)
M
i
N
j
ij
ij
y
x
MN
MSE
1
1
2
1
MSE
L
PSNR
20
log
10
10
Non uniform leads to intervals of different sizes
Adaptation to distribution of values to quantize
Optimal Quantizer
Objective:
Find best d
iand q
i
hypothesis:
optimisation criterion
probability density p(x)
Criterion of Mean Square Error
Minimization of MSE
non uniform probability density
non linear
quantization
Non uniform Quantization
Quantization CH 4
11
Non uniform Quantization
MSE Minimization – MAX Quantizer
reconstruction levels: centroïds of areas defined by
p(x) and decision regions
decision thresholds d
i: in the middle of limit values of
intervals
Symetry with 0
Impossible d’afficher l’image.
1
2
,...,
2
,
1
0
)
(
)
(
i i d d iL
i
dx
x
p
q
x
2
1
2
,...,
2
,
1
2
0
0
1L
i
L
i
q
q
i
d
i i i i i i id
q
q
d
et
12
Outlines
Definition
Scalar quantization
Definition
Distortion
Non uniform quantization
Quantization CH 4
13
Principe / example
4 color Image
Symbols 00 / 01 / 11 / 10
Vector Quantization
00 00 00 01 01
00 01 10 11 01
00 01 10 11 11
01 10 11 11
4 color Image Image:
Symbols 0/1
1 1
1 0
Dictionnary
Binary
Q = 2
0
1
QUANTIZATION
VECTOR
Index choice for
the nearest one of
current region
14
Resulting number of bits
Let:
Image : matrix MxN
I(x,y)
[ L
min, L
max]
Necessary number of bits for representation of grey
levels in L is K
So for scalar quantization:
L = 2
K
Total number of bits:
b = M
x
N
x
K
Then for vector quantization:
blocks
m
x
n
p blocks to code
M
: dictionary size
Quantization CH 4
15
Bibliographie
[1] Véronique Coat, Cours et supports de cours, INSA Rennes
[2] Max Mignotte, "Traitement d'images – Introduction", support de
cours, Université de Montréal
[3] Grégory Bizarri, "Etude des mécanismes de dégradation du
luminophore", thèse de doctorat, décembre 2003
[4] Sofiane Lariani,
"
Perception et interprétation de sections et blocs
sismiques: oculométrie et analyse d'images", Thèse de l'UJF,
Mathématiques Appliquées, Grenoble, 4 Octobre 2000.
[5]
h
ttp://www.chusa.jussieu.fr/pedagogie/pcem1/biophysique/opt_phys_A_2005.pdf
[6]
Adelson, « E.H. Lightness Perception and Lightness Illusions ». In
The New Cognitive Neurosciences,
2nd ed., M. Gazzaniga, ed.
Cambridge, MA: MIT Press, pp. 339-351, (2000).
[7] Pierre Kornprobst, Cours et supports de cours, INRIA
[8] M. Burel, C. Obert, « DICOM – Quantification vectorielle », 2005
[9] Pierre MATHIEU, Cours, DEA ARAVIS, Polytech’Nice-Sophia
16
MAX Quantizer
Uniforme Gaussien Laplacien Rayleigh
bits di ri di ri di ri di ri 1 -1.0000 -0.5000 - -0.7979 - -0.7071 0.0000 1.2657 0.0000 0.5000 0.0000 0.7979 0.0000 0.7071 2.0985 2.9313 1.0000 - - - 2 -1.0000 -0.7500 - -1.5104 - -1.8340 0.0000 0.8079 -0.5000 -0.2500 -0.9816 -0.4528 -1.1269 -0.4198 1.2545 1.7010 -0.0000 0.2500 0.0000 0.4528 0.0000 0.4198 2.1667 2.6325 0.5000 0.7500 0.9816 1.5104 1.1269 1.8340 3.2465 3.8604 1.0000 3 -1.0000 -0.8750 - -2.1519 - -3.0867 0.0000 0.5016 -0.7500 -0.6250 -1.7479 -1.3439 -2.3796 -1.6725 0.7619 1.0222 -0.5000 -0.3750 -1.0500 -0.7560 -1.2527 -0.8330 1.2594 1.4966 -0.2500 -0.1250 -0.5005 -0.2451 -0.5332 -0.2334 1.7327 1.9688 0.0000 0.1250 0.0000 0.2451 0.0000 0.2334 2.2182 2.4675 0.2500 0.3750 0.5005 0.7560 0.5332 0.8330 2.7476 3.0277 0.5000 0.6250 1.0500 1.3439 1.2527 1.6725 3.3707 3.7137
Decision thresholds and reconstruction
Levels for MAX’s quantizer
Quantization CH 4
17
Uniforme Gaussien Laplacien Rayleigh
bits di ri di ri di ri di ri 4 -1.0000 -0.9375 - -2.7326 - -4.4311 0.0000 0.3057 -0.8750 -0.8125 -2.4008 -2.0690 -3.7240 -3.0169 0.4606 0.6156 -0.7500 -0.6875 -1.8435 -1.6180 -2.5971 -2.1773 0.7509 0.8863 -0.6250 -0.5625 -1.4371 -1.2562 -1.8776 -1.5778 1.0130 1.1397 -0.3750 -0.3125 -0.7995 -0.6568 -0.9198 -0.7287 1.5064 1.6277 -0.2500 -0.1875 -0.5224 -0.3880 -0.5667 -0.4048 1.7499 1.8721 -0.1250 -0.0625 -0.2582 -0.1284 -0.2664 -0.1240 1.9970 2.1220 0.0000 0.0625 0.0000 0.1284 0.0000 0.1240 2.2517 2.3814 0.1250 0.1875 0.2582 0.3880 0.2644 0.4048 2.5182 2.6550 0.2500 0.3125 0.5224 0.6568 0.5667 0.7257 2.8021 2.9492 0.3750 0.4375 0.7995 0.9423 0.9198 1.1110 3.1110 3.2779 0.5000 0.5625 1.0993 1.2562 1.3444 1.5778 3.4566 3.6403 0.6250 0.6875 1.4371 1.6180 1.8776 2.1773 3.8588 4.0772 0.7500 0.8125 1.8435 2.0690 2.2971 3.0169 4.3579 4.6385 0.8750 0.9375 2.4008 2.7326 3.7240 4.4311 5.0649 5.4913 1.0000
MAX Quantizer
18
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