Variable Metric For Binary Vector Quantization UNIVERSITY OF JOENSUU DEPARTMENT OF COMPUTER SCIENCE...

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