, ,

56

A. A. Novosyolov

PARAMETRIZATION OF MODELS OF CONTROLLED SYSTEMS The paper describes application of orthogonal series method for construction of controlled systems models under

non-parametric uncertainty. A key element of the method is draw of orthogonal expansion length based on observa-tions, in other words, defining parametric structure of the model. The method is demonstrated for estimation of distribu-tion density and regression function. Directions for generalizing onto multi-dimensional case are also presented.

Keywords: distribution density, regression function, orthogonal series, non-parametric estimate.

. ., 2010

004.932.2 . . , . . , . .

*

; . - .

: , -

, .

[1]. - , - . , () [2; 3], - .

- [4; 5]. , , - - - -. - , - . - , - .

[68], - . ,

, - . . [9].

-, [10; 11]. - - MeanSC ( [12]) .h . - , - , - .

- . , k- , - N , -

( ) ( ) ( )1{ ( , ..., ) , 1, },

i i i kkX x x x R i N

( )ijx -

i- j- -

( 1, ).j k

* ( 09-07-

12087-_).

. .

57

( )ix k- x, - ( ),f x 1, , kkx x x R -- - . - ( )f x kx R - ( ),Nf x

( )

1

1 ( ) ,iN

N ki

x xf x

hNh

h ; ( )x - (), - [10; 13].

, , - ,

2( ) || || ,kx c x

0;kc : [0; ) R , :

( ) 0;t 1 2( ) ( ),t t 1 2 ;t t 0

( ) .t dt

, , - [13]:

21

E

1( 2)(1 ), 1,

20, ,

kV k x x

kV k- - ;

2|| ||/ 2 2

N ( ) (2 .)x

kx e

- ,

( )f x ( )f x ([14])

2

1

( ) ;N

k iN k

i

c x xf x

hNh

2

21

2 ( ) ( ) ( ) .N

k iN ik

i

c x xf x f x x x

hNh

( ) ( ) ( ) / ,t t d t dt ,

( )t [0; )t .

, [15; 16]

2

1

2

1

( )

Ni

ii

hN

i

i

x xx

hm x x

x xh

.

, - ( )f x x.

, - 0

kx R 10 0 0( ),hx x m x 10x

2 1 10 0 0( )hx x m x . . 0 ,x

0( ) 0,hm x .

[14], - ()

( ).f x , 0x 0 ,x

-

0 0, , .x x

- X : ix jx ,

, , - . , . [17; 18] , - .

- MeanSC ( - [12]), , -. ( )hm x -

( )Nf x - .

MeanSC. - . - ( 0 1,K K , 256), - . - - , - , - - . MeanSC h, , T -.

1. kx R

( ).Nf x

kx R ,x

, x, .x -

a , -

, ,

58

, - a.

2. , - x ( ( ) ),Nf x

-

( ) ,Q x X

.x x X ,

0:x

0 ( ) .Nf x

0 . 3. -

,C X , - :

1) x C ( ) ;Q x 2) 1 2,x x

, 1 2( ), ( )Q x Q x

,kP R 1x

2 ,x

1 2 ( )

. min ( ), ( )

N

N N

f xT

f x f x

1T , - .

, - MeanSC .

1. - . - - 1[0; 1] [0; 1]kK K 2h (h ). ( ) - X.

2. - X. x x X. - - ( ) - -.

3. () S . , X, , . S.

4. s S s

( ),Nf x s.

0 | , ( ) .NZ s s S f s 0 , , i

s S

S s s s

1 1( ), , ,i h is m s s ,

.

5. x X ,S . X - 2.

6. . ,P

1( )Q x 2( )Q x

( 3), , , ,

1( )Q x 2( ).Q x

-

1 1( )x Q x 2 2( ),x Q x

, - 3

1 1 1 2 2 2, , , , , .P x x x x x x 1 2,x x , 1x 2 .x , x P - 1 2;x x . h.

, - , - , -, x. .

. - , - MeanSC, - h. [68], - -. . , [7] - , - . - , - ( -) . ; - L , - MeanSC h.

.

- ( ), ( ), - (1) ( ) ( ){ , , , , },l LG G G ( )lG l- - , ( )lM .

. .

59

( )lH

,( ) ( )l i j lH H ,N N - l- , :

,

0, ,

1, , )

(

i j lH

, 1, , ,i j N .i j

L

, ,i jH H , ,1

1( ),

L

i j i j ll

HL

H

, 1, ,i j N . ,i jH -

ix jx -

. -, - . - , .

-, - , [9]. , . . , . - , .dT , , - .

.

, ( ) U , 2M . - - ( | ) ( ),rp x U r f x 1, , .r M -. , , ( ),rP U r 1, , ,r M

1

1.M

rr

P

( )rf x x. .

- - X M . - , - , . . ,

. a b :

,

0, ,

1, , ( )

a b

H

, ,a b X .a b a b -

. ( ( ) ( ))UP U a U b -

. , 2M

22

1

1 ( ( ) 1 | ) ( ( ) 1 | ) ( ( )

( ) ( )2 | ) ( ( ) 2 | ) 1 ,

( ) ( )r

r

r r

UP U a a U b b U a

f a f b Pa U b b

p a p b

2

1

( ) ( ) ,rr

rp f P

, .a b

, a b

er ( ),P

,er

,

, ( ) 0,( )

1 , ( ) 1.U a b

U a b

PP

P

H

H

,

er , ,

,

( ) (1 ( )) ( )(1 )

(1 2 ) ( ).a b U a b U

U U a b

P P P

P P

H H

H

- : ( ). - , - , ,( ( )) ( ),a b a b H H

er er( ( )) ( ).P P L- -

- 1, , L

1( ), , ( ).LH H , , L . () - -

1 1

1 10, ( ) ,

( ( ), , ( )) 2

1, .

L

lL l

HH H L

H

- [6].

1. -

( ) :

er ( ( ) ,) 1 2U U HP P P P 2

erVar ( ) (1 2 ) (1 ),U H HP P P P

( ( ) 1).HP H

, ,

60

er 1( , , )LP -, , , - - a b. 1, , L - . .

2. - - :

1 , , er 1 ,( , , ) (1 2 ,)

L L U U LP P P P H

1

2, , er 1 , ,Var ( , , ) (1 2 ) (1 ),L L U L LP P P P H H

,1

12

1 1( ) (1 ,)

2

L Ll l L l

L l L H HLl

l

P H C P PL

H

. -

. -, - er ( ) 1/ 2,P . . , - -, . 1 , : 1/ 2HP 1/ 2;UP

1/ 2HP 1/ 2.UP , -, .

3. er ( ) 1/ 2P 1/ 2HP 1/ 2,UP

- , 1 ,UP .

, - .

-. , - C++

1EMeanSC { , , }, , , .L dh h h T T 1, , Lh h ; ; T , dT .

- -. 1 2 - . 3 -. 4 - . 5 dT . 1. , 400 , - (. 1), (. 1, ). - PRTools (PRTools: the Matlab Toolbox for Pattern Recognition http://www.prtools.org). , - , - . 1, . 2 , - 200 .

2. - (. 2), 200 , - 100 (. 2, ). , - . . 2, - MeanSC - . , . -, -, , (. 2, ).

0 20 40 60 80 100 120 140 160 180 200 220 24020

40

60

80

100

120

140

160

180

200

220

0 20 40 60 80 100 120 140 160 180 200 220 2402040

60

80

100

120

140

160

180

200

220

. 1. , 400 : ; EMeanSC

{10;10,5;11;11,5;12;12,5},h 0 , 4,T Td = 0,99 (2 )

. .

61

20 40 60 80 100 120 140 160 180 200 22020

40

60

80

100

120

140

160

180

200

220

20 40 60 80 100 120 140 160 180 200 22020

40

60

80

100

120

140

160

180

200

220

20 40 60 80 100 120 140 160 180 200 22020

40

60

80

100

120

140

160

180

200

220

20 40 60 80 100 120 140 160 180 200 22020

40

60

80

100

120

140

160

180

200

220

20 40 60 80 100 120 140 160 180 200 22020

40

60

80

100

120

140

160

180

200

220

. 2. , 200 : ; MeanSC (12, 14 12 )

{8; 8,5; 9},h 0, 5,5;T EMeanSC

{8; 8,5; 9},h 0, 5,5,T 0,999dT (2 )

3. -

, 3 000 (. 3), - 3 (. 3, ). , - , ( ) (. 3, ). - (. 3, ) .

4. (- 500 450), ALOS/ANVIR-2 17 2007 . (. 4). - 5552'14.8'' 5555'2.13'' - 8350'45.41'' 8354'51.6'' . . 4, . -. - 2,4 ( 2 ). - - 18 . - (. 4, ).

5. - (. 5, ) 510 604 . -

.R G B -

. , EMeanSC, , ,dT . 5, . , - -.

- - , - . . EMeanSC, -. , - - -.

, - , - .

, ,

62

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

. 3. , 3 000 : ; 3 5 (19, 8 5 , {10; 15; 20}h );

EMeanSC {5; 10; 15; 20; 25},h 0, 2,5,T 0,75dT (3 )

. 4. : ALOS; EMeanSC {5; 10; 15},h 0, 1,3,T 0,5.dT

6 : 1 ; 2 - ; 3 ; 4 - ; 5

. .

63

. 5. : ; , , EMeanSC {5; 7; 9},h 0,

1,5,T 0, 4dT (18 ), 0,6 (12 ) 0,95 (6 )

1. ., . -. . : , 2006. . 812.

2. Dey V., Zhang Y., Zhong M. A review on image segmentation techniques with remote sensing perspective // ISPRS TC VII Symposium 100 Years ISPRS, Vienna. Austria. July 57 2010. IAPRS. Vol. XXXVIII. Part 7A. P. 3142.

3. Rekik A., Zribi M., Hamida A., Benjelloun1 M. Review of satellite image segmentation for an optimal fu-sion system based on the edge and region approaches // IJCSNS International Journal of Computer Science and Network 242 Security. 2007. Vol. 7. 10. P. 242250.

4. Jain A. K., Duin R. P. W., Mao J. Statistical Pattern Recognition: A Review // IEEE Transactions on Pattern Analysis and Machine Intelligence. 2000. Vol. 22. 1. P. 437.

5. Clausi D. A. K-means Iterative Fisher (KIF) unsu-pervised clustering algorithm applied to image texture segmentation // Pattern Recognition. 2002. Vol. 35. 9. P. 19591972.

6. . . // . 2010. . 15. 1. . 4052.

7. Strehl A., Ghosh J. Clustering ensembles a knowledge reuse framework for combining multiple parti-tions // The Journal of Machine Learning Research. 2002. Vol. 38. P. 583617.

8. Hong Y., Kwong S. To combine steady-state genetic algorithm and ensemble learning for data cluster-ing // Pattern Recognition Letters. 2008. Vol. 29(9). P. 14161423.

9. Jain A. K. Data clustering: 50 years beyond K-means // Pattern Recognition Letters. 2010. Vol. 31, Is. 8. P. 651666.

10. Parzen E. On the estimation of a probability den-sity function and the mode // The Annals of Mathematical Statistics. 1962. Vol. 33. P. 10651076.

11. Rosenblatt M. Remarks on some nonparametric estimates of a density function // The Annals of Mathe-matical Statistics. 1956. Vol. 27. P. 832837.

12. . ., . . - -

, ,

64

grid- // -. 2006. . 42. 2. . 9099.

13. . . // - . 1969. . 14. 1. . 156160.

14. Comaniciu D., Meer P. Mean shift: A Robust Ap-proach toward Feature Space Analysis // IEEE Transac-tions on Pattern Analysis Machine Intelligence. 2002. Vol. 24. 5. P. 603619.

15. Fukunaga K., Hosteeler L. D. The estimation of the gradient of a density function, with applications in

patter recognition // IEEE Trasactions on Informational Theory. 1975. Vol. 21. P. 3240.

16. Cheng Y. Mean shift, mode seeking, and cluster-ing // IEEE Tans. Pattern Analysis and Machine Intelli-gence. 1995. Vol. 17. P. 790799.

17. Comaniciu D., Meer P. Distribution Free Decom-position of Multivariate Data // Pattern Analysis and Ap-plications. 1999. Vol. 2. P. 2230.

18. Freedman D., Kisilev P. Fast Mean Shift by Com-pact Density Representation // IEEE Conference on Com-puter Vision and Pattern Recognition. 2009. P. 18181825.

I. A. Pestunov, V. B. Berikov, Yu. N. Sinyavskiy

ALGORITHM FOR MULTISPECTRAL IMAGE SEGMENTATION BASED

ON ENSEMBLE OF NONPARAMETRIC CLUSTERING ALGORITHMS

The method for constructing an ensemble of nonparametric clustering algorithms is proposed. Its theoretical sub-stantiation is resulted. Results of the model data and real images confirm the efficiency of the proposed method.

Keywords: multispectral image segmentation, nonparametric clustering algorithms, ensemble approach.

. ., . ., . ., 2010

519.24

. .

, , . - .

: , , , , -

, , . -

: - .

[1]. , - -. - - .

, , . , , .

- . . [1] : - , , 6 :

- ;

;

, - ;

, - , - ;

- -;

, - - .

--: -.

, [1]. ,