Variable Metric For Binary Vector Quantization UNIVERSITY OF JOENSUU DEPARTMENT OF COMPUTER SCIENCE JOENSUU, FINLAND Ismo Kärkkäinen and Pasi Fränti
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Variable Metric For Binary Vector Quantization
UNIVERSITY OF JOENSUU DEPARTMENT OF COMPUTER SCIENCEJOENSUU, FINLAND
Ismo Kärkkäinen and Pasi Fränti
Distance and distortion functions
Distance function:
Distortion function:
dK
k
d
jkikjid cxcxL
1
1
,
N
ipidN i
cxLCXE1
1 ,,
Distortion for binary data
Internal distortion for one variable:
djkjkdjkjkjk crcqD 1
qjk = the number of zeroes rjk = the number of onesCjk = the current centroid value for
variable k of group j.
Optimal centroid positionOptimal centroid position depends on the metric. Given:
1/1 dThe optimal position is:
jkjk
jkjk
rq
rc
Proposed algorithm
Create initial solution. Set d to dmax. While d > dmin, do: Perform one GLA iteration. Decrease d. Set d to dmin. Iterate GLA until solution has converged. Return final solution.
Example of centroid location
0.50
0.60
0.70
0.80
0.90
1.00
6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0
d
c
q=10,r=90
q=20,r=80
q=30,r=70
q=40,r=60
Second example
q = 7 5 r = 2 50 .0 1 .0
c = 0 .5
c = 0 .3 7
c = 0 .2 5
c = 0
d = ( = 0 )
d = 3 ( = 0 .5 )
d = 2 ( = 1 )
d = 1
Test images
Blockwise quantization of pixels into two levels according to the mean value of the 4x4 blocks.
44 pixel blocks.
Results for Bridge
Bridge1
.48
9
1.4
67
1.3
87
1.4
41
1.3
86
1.2
93
1.3
67
1.3
32
1.3
08
1.2
45
1.1
1.2
1.3
1.4
1.5
1.6
GLA SA Split PNN
Dis
tort
ion
Binary
Soft
Prop.
Results for Camera
Camera1
.77
1
1.7
65
1.6
74
1.6
93
1.6
46
1.5
45
1.6
59
1.6
01
1.5
56
1.4
96
1.4
1.5
1.6
1.7
1.8
GLA SA Split PNN
Dis
tort
ion
Binary
Soft
Prop.
Results for CCITT-5
CCITT-5
0.0
33
0.0
40
0.1
37
0.0
26
0.0
31
0.0
29
0.1
65
0.0
26
0.0
45
0.0
27
0.00
0.05
0.10
0.15
0.20
GLA SA Split PNN
Dis
tort
ion
Binary
Soft
Prop.
Results for DNA
DNA7
.05
6
6.6
22
6.5
74
7.2
30
6.7
57
6.6
62
6.5
96
6.5
70
6.6
27
6.5
70
6.2
6.4
6.6
6.8
7.0
7.2
7.4
GLA SA Split PNN
Dis
tort
ion
Binary
Soft
Prop.
Different parameter combinations (Bridge)
GLA SA
Linear Exponential Exponential
d0 0.05 0.1 0.25 0.95 0.99 0.95 0.99 2 1.325 1.335 1.359 1.318 1.314 1.278 1.254 3 1.319 1.327 1.346 1.312 1.310 1.269 1.245 4 1.318 1.325 1.344 1.310 1.309 1.268 1.246 5 1.317 1.324 1.344 1.311 1.309 1.270 1.249 6 1.318 1.323 1.342 1.311 1.307 1.271 1.252
Different parameter combinations (DNA)
GLA SA Linear Exponential Exponential
d0 0.05 0.1 0.25 0.95 0.99 0.95 0.99 2 6.679 6.718 6.728 6.646 6.668 6.571 6.571 3 6.646 6.688 6.701 6.628 6.641 6.570 6.570 4 6.685 6.644 6.647 6.643 6.641 6.570 6.570 5 6.662 6.661 6.671 6.630 6.627 6.570 6.570 6 6.663 6.643 6.659 6.637 6.682 6.570 6.570
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