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