Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://imaging.cs.msu.ru/pub/2007.JProg.Yurin_Volegov.Draft.ru.pdf Äàòà èçìåíåíèÿ: Mon Oct 13 23:00:00 2008 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 19:46:44 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 63

,
.. *, .. ** * - ( ) ** . .. dvolegov@rambler.ru yurin_d@inbox.ru . . 6 . . , . . ( ), . , 3D , . : , , , , , , , , .
1.

, . [1], . , .. , [1,12]: 1) , ; 2) , ; 3) , , , ; . , . , . : , , .


() [2-4]. [2] () , . 1.8 [2]. - , . , . , , . , .. . , , , , - , , , . , / , . [3] (Image Registration, Image Matching), . . [2] , , 4 . , [3] , , . [2] , . . [5], [3] . [4] , 6 , , , . . [6], , . (Scale Space) [7]. 7, [8,9]. RANSAC [10] , . [4] RANSAC , 50%. [4] , . , , , , , , . [11], [12] [13-21]. , .


[11] , , , . (, ) - [2]. . . [1] ( 0 N ), , . ( ), [2]. , , . , , ( ), [22]. , [2] 1/3 45°. [2] , , [4] [3] , , , RANSAC [10] , . , [11] , . , , , . , [11] ( ) (. 4.4). [12] . [4]. , , , , . , . [11] . [6,4,7], . RANSAC 4 . , . , -


. . . [11] , , , , . , [12] «» . , , , . , , [13-15] [16-21], (shape from motion) . , , , [1], (Rectified stereo). , [23] . , (disparity map), . [13] , . , , , . [23] , 7 . , [24,25], , , , , RANSAC. , , . ( ) [21] , [16-20]. , , , . , , / . , , , , ,


Loreo (www.loreo.com), , - , . [16-20], , , , . , . , [6,4] [7] [8,9]. . , . [26] , . [7,4], [27,28] , . , . , , . . [27] . , , - . , , 12 . RANSAC . , , . [29] , , , . . , . , . , . , , ( , [30]). , . , , . ,


. . , [29] , , . , . , , , , , . , . . [31], (), () , - . , , .
2.

[32]. , , , . , . , . , , . , , [44], [45], [46]. , . N 2 N ( N - 1) / 2 . O ( N ) , ( N ~ 10 ). [33]. . 2.4. O ( N log N ) [34,35]. [34] . kN , k = 2 Â 5 . [35] . , [34]. , , , , . [36,37] . 6


. N O ( N log N ) '' , , O (log N ) '' . , N 16 N

log N + 1 , 2

( , [34,35]). [35], 2.4, 4.3 2.1 , [38]. cas :

cas( x) sin(x) + cos(x) = 2 sin( x +


4
n

)

(1)

H ( ) f ( x) , x, R [38]:

r

r

rr

r r rr r H ( ) = f (x)cas(2x T )dx r r rr r f (x) = H ( )cas(2x T )d

(2)

:

(3)

:

r r H ( ) = Re( F ( )) + Im(F r r F ( ) f ( x) : r r rr F ( ) = f (x)exp(2ix T
:

r ( )) r )dx

(4)

(5)

r r rr r f (x) = F ( )exp(-2ix T )d

(4) (5). (5) [38] [39], [39]. h = ( h0 ,L h () f = ( f 0 ,L f

r

r

N -1

) T

N -1

) T [38]:
hi = 1 N


j =0

N -1

f j cas

2ij , i = 0K N -1 N

fj =


i =0

N -1

2ij ji cas , N

(6)

j = 0K N - 1

[38]. () O ( N log N ) .


[38] N â N F N â N H :

H


i, j

=


k =0 l =0

N -1 N -1

Fk ,l cas

2 (ik + jl ) , i, j = 0K N - 1 N


(7)

( cas

2 (ik + jl ) 2ik 2jl ). [47], cas cas N N N





, (10) (9). (1) sin() cos() :

cas(a + b) =

1 [casacasb + cas(-a)casb + casacas(-b) - cas(-a)cas(-b) 2 H 1 [H i, j + H p, j + H i,q - H p ,q ] 4 p = ( N - i ) mod N , q = ( N - j ) mod N
i, j

]

(8)

(8), (7)

=

(9)

H 'i , j =


k =0

N -1

cas

2ik N


l =0

N -1

Fk ,l cas

2jl N

(10)

a mod b a b .

H

0, 0

(7) ( ) ( ).

. F H (. 1):

H

0 i, j

=H

r ,s

r = (i +

N N ) mod N , s = ( j + ) mod N 2 2
0

(11)

H i , j . (11) (9) (. 1):

H

0 i, j

=

H r,s + H ,s + H r,q - H p p 4

,q

N N - i ) mod N , q = ( - j ) mod N 2 2 N N r = ( + i ) mod N , s = ( + j ) mod N 2 2 p=(

(12)

, , (12) , inplace ­ (9),(10) . , . , log N


; , , .

. 1.

. 1. (12)

1. F (inplace). 1. (. 1); 2. F (inplace); 3. (inplace); 4. (inplace) (.. ); 5. (inplace); H ' ; 6. H ' (9),(10), H F ( N / 2, N / 2) . 2.2 . . , (). [40]. :
0

( x = x + ( x - u 0 )(k1r 2 + k 2 r 4 ) ( y = y + ( y - v0 )(k1 r 2 + k 2 r 4 ) ,
2

(13)

r 2 = ( x - u 0 ) 2 + ( y - v0 )

x, y ( ), x , y . u 0 , v0 , k1 , k 2 . [41]. x, y B 2 â 2 :

((

B ( x, y ) =


i

I ( x, y ) rr r g igT , gT = i i i x

I i ( x, y ) y

(14)


I i ( x, y ) ( RGB). =2Â3 . B C ( x, y ) x, y :

r r C ( x, y ) = 1 (B( x, y )) - 2 (B( x, y )), Bh i = i h i , 1

2

(15)

C ( x, y ) (15) . C ( x, y ) . 2.3 . 2.3.1 , - : , , (. 2). . [0, ) . . , :
. 2.

x cos + y sin =

(16)

r 2.3.2 m -

x, y .
r

m , ,r (. 3). m :

r m=

cos sin 2 2 w + w

-

r , | m |= 1 w



T

(17)

w , /. 2.4 ( ) . , [34] , 6 . I ( x, y ) ( ). , . I ( x, y ) [33]:

R ( , ) =



I ( x, y ) ( - x cos - y sin )dxdy

(18)




( x)dx = 1

(19)

( x) - . R( , ) I ( x, y ) , . , , , R ( , ) . [33] , . x, R

rr

2

.

:
r r A f ( x ) = g ( )

(20)

:
: : :

H

(1)

r r f (x) (1 , x2 ) = f (x)cas(2x11 )dx1
( 2)

(21)

H

rr r rr r f (x) ( ) = f (x)cas(2 T x)dx

(22) (23)

:

r P f (x) (r , ) = f (r cos , r sin ) r r rr r R f (x) ( , ) = f (x) ( - x T k )dx, r k (cos , sin ) T

(24)

1. r f ( x) :
r R f ( x) ( , ) = H (1)

PH

(2)

r f ( x ) ( , ) ,
(1)

(25)

. (3), H

(25) : (26)

H H
(1)

(1)

r R f ( x ) ( r , ) = P H

(2)

r f ( x) ( r , )

(19),(24),(21) (26) :

r r rr r r rr r R f (x) (r , ) = f (x) ( - x T k )cas(2r )dxd = f (x)cas(2rx T k )dx
( 2) r r rr r r r rr r f (x) (r , ) = P f (x)cas(2x T ))dx ( ) = f (x)cas(2rx T k ))dx

(27)

(26):

PH

(

)

(28)

, (27) (28) . (25) I ( x, y ) :


2. . 1. H ( x, y ) I ( x, y ) (. 1).

2. H ( x, y ) , P ( r , ) . ( r , ) P ( r , ) H ( x, y ) ( r cos , r sin ) ( ). 3. P ( r , ) . 1, R ( , ) I ( x, y ) . ( ). , [34]. .3 (. 3) (), . .3. , (. 3), . 3 ­ (. 3), , 2.2. , . . . (. 3,), . . 3,. . 3, ( ), . , , . , , (. 2.2) . . 3,. , ( ) .

d2 L( x) = 2 dx
:

x2 exp(- 2 ) 2 2 2 1

(29)

(H

(1)

L( x))( ) = 2 exp(-

2
2

2

)

(30)

[38], ( ), ( ). L( x) , (25) :
r R f ( x ) ( , ) = H (1)

FL PH



( 2)

r f ( x ) ( , ) ,

(31)

FL L( x ) .


(25) , u1 ( x) -

u 2 ( x, y ) = u1 ( x 2 + y 2 ) .

. 3. (1224x1632) . 3. (1224x1632)

. 3. (1224x1632)

. 3. . 3 (2048x2048)

. 3. . 3 (2048x2048) . 3. . 3.

. 3. . 3 (2048x2048) . 3. . 3 (2048x2048) . 3. . 3.


[34] , . , . . [41], (. 2.2), , . [34] . , , , 2 ( , [34]) , (, ..). , , . . 2, . , . [41] . . . «» . «» ( ), . 3. . 1 1. I th : I th = I min Lmin , I min Lmin 3 , . . 2. . , «» , , .

3. , B : a. B , , «» . b. B . 4. , .2. 5. , 4 , - , . 6. , «» .
2

, , , . (.4) . Intel Pentium 4,


2.8 , 1 , 1 Windows-XP, Microsoft Visual Studio 2003 (ver. 7.1). 100 10 [34] 1 81928192 . 0.1 . Proposed algorithm 0.01 Algorithm [34] 64 128 256 512 1024 2048 4096 8192 [34]. Image width (=height), pixels . , - . 4. 2 O( N log N ), N = M . , [34] 6 , [39] , [38], [38], ( , Intel P3, P4 , [38]). (128 256 ) 6 1.5 , . 8192 6 - , [34], . , [34] 6 , . , [34], . , AMD : 8 6.
Calculation time, seconds

3. ()

, P ( 3x3), , [1]:

r r h1 h 2 - :

r r h 2 = P h1 ,

(32)

r h i = ( ~i , ~i , ~i ) T , i = 1,2 xyz

(33)


, P -, . 3.1 (. 5). r F1 F2 . n . i1 , j1 , k 1 i2 , j2 , k 2 .

rrr

rrr

1 2 , k 1 k 2 .
r rrr rrr R ( i1 j1 k 1 ) = ( i2 j2 k 2 )

r

r

1 2 . i1 , j1 i2 , j2

rr

rr

t . R : (34)



F1 d1 , ( d1 0 ).
, 1 2 :

r rT t P = R I - n d1
T

(35)

. 5. .

I - 3x3.

3.2 P . 1 , 1 2 , 2 - r . m - (. 2.3.2). , i - : rr (m i , h i ) = 0 , i = 1,2 . (36)

, (32) :

r m 1 r m2 = r, | m 1 |

(37)



= ( P -1 )

T

(38)

- [39]:

r A -1u r r T -1 -1 ( A + uv ) = A - 1+

r ( vA -1 ) rr vA -1u

T

(39)


(35) P , (38) :

rr nt T = R I + rr d1 - ( t , n)
T

(40)

2. , , :

r | t | d1 ,

(41)

:

1 1 | ( ) | 1+ 1-
. r , e :

(42)

rr r n t T r n r T | e |= R I + r r e = I + d1 - ( t , n) d1 - rr n t T r 1 r r e 1 =1- I + 1+ 1+ d1 - ( t , n)

rT r t r r e ( t , n) 1 + = 1- 1-



(43)

r t , . (42)
4.1.1, 4.2.
4.

.

P , 1,i , 1,i 2 , j , 2 , j . : 1. . 2. . 3. , , . . ( ). , . .


4.1 4.1.1 F , :

F = F (P, 1,i ,1,i , 2, j , 2 , j )

(44)

, P , ( ) P . :

F=


i =1 j =1

N

1

N

2

cij Fij

(45)

r r | m 2 , j â m1,i |2 Fij = - exp - 2 2 f

(46)

cij i - j - , 4.3, 4.4 Fij i - ( P ) j - . i - P j - , Fij -1.

f ,
r |m r â m1,i |>>

. (42) r r , m1,i m 2 , j f , | Fij |<< 1:
2, j
f

| Fij | 0

(47)



f = 0.01 Â 0.2 ,

f (. 4.2). f : f <
1 (48) min sin j , j , 2 j ,j ,j j r m 2 , j . (48) ,
1 2 1 2 1 2 2







j1 , j

2

- m

r

2 , j1

i Fij .
, .
4.1.2 (45) (. r (35)): R , n ,


r t . , d1
3x3. , , , , . , . n k 1 , , , . r k1 :

r

r

r n = (0,0,1)

T

(49)

r k 1 (. 6, ) . , i1 r r r j1 , j1 i1 . R :
. 6.

r

cos R = R = sin 0

- sin cos 0

0 0 1



(50)

(49) (50) :

r :

r ( , t ) =

cos - sin

sin cos



1-

x



y z

z

1-

0 0 1 1- z

(51)

rt = d1

(52)

(44) r ( , ) , (51). , (45) , , , . . , (. 4.2) , . -


, , .

4. : r 1. ( , ) . 2. (10~100), . 3. [39]. 4. , . . , «» . , ( 16 !). . r m . :

r rr r f 0 ( , t ) =| m â ( , t )m |

2

(53)

:

r F0 (0, 0) . 0 r , , t . , r ( ) F0 (0, 0) :
2 2 2 ( mx + m y ) 0 mx H0 = 0 mx m 2 0 mx m

r r f 0 ( , t ) F0 ( , t ) = - exp - 2 2

(54)

1

0

0 mx my
y z

mm

m my mz
2 y

mm m

x z y z 2 z

0

(55)

H 0 :

1 = 2 =

2 (m x2 + m y )



3, 4

2

r h1 = (1,0,0,0)
3, 4

T

r 0 h2 = r m

=0

r h

= , 0 r v

(56)

rr v m

, F0 r m , r , , m . r | m |= 1 , (56) :


max( (H 0 ))

1



2

(57)

N1 (48) :

max( (H 0 ))

N



1 2

(58)

(48) Fij , i j . (58), , . , , :

1r r r r F ( s ) = - N c + s T Hs , s 2

=r t

(59)

r max( (H )) 1 2 , , | s |< r : | F ( s ) | N c / 2 .

N c - .

| F | N c / 2 . N c ,

, .1 4 , .2 ,

N c = min( N1 , N 2 ) / 2 . ,

N c , .
4.2 , , . , f ,

, . f , . r , i n i di . i . . r R t . i i , i , :

rr ni t T rr i = R I + di - (t ,ni )
T

(60)


m

r

1,ij

,

i r m 2 ,ij . (37) r i m1,ij r = m 2 ,ij , i, j r | i m1,ij |

j - ,

. -

(61)

( , ) n 0 d 0 , 0 :

r

rr n0 t T rr 0 = RT I + d0 - (t,n0 )

(62)

q ( 0 ) :

q ( 0 ) = max i , j , i, j r r i , j - m 2 ,ij 0 m1,ij .
m

(63)

(62) 0 ,

r

1,ij

0 m

r

1,ij

m

r

2 ,ij

-

q ( 0 ) . (48),

f q( 0 ) -

. r q ( 0 ) , n i di . , q ( 0 ) n i di . ( ) , , r 1. n i (

r

r n i , , .

r |t| 2. [0, ] di
q ( 0 ) :

. 3)

q( 0 )


1-

(64)

(48) (64)

f:
(65)


1-


f

1 min sin 2 j ,j ,j j
1 2 1 2

j1 , j

2



j1 , j

2

(48) .



f

, 4 . f (65).


: . . , , , (



j1 , j

2

<



1-

)

. , . f . (32).
4.3 ­ , . , , . . , . , , . , (/), ( ). [42,43]. «» . (), . , . cij (45). l

d 1 d 2 ­ . d : r , g , b :

r

r

r

r d = ( r , g , b)

(66)

, r , g , b [0,1] . r I (p ) O (. 7). C (p ) . O : 1. O 2 wm = 6 , l . , (. 2.2). r 2. O I (p ) .

r

. 7. ,


q O wq , . -, q p , l , r , q . , l , , , · · ; , .

r

r

r

I (p ) = ( R (p ), G (p ), B (p )) , r p = ( x, y ) . :
T

rr

r

r

r

4.

p l :

r d = 0, W =0
o

r

r w p = C (p ) r o m l, p m .

r r q0 = p : r · qn =

·

r rr n -1 ) | p - q n |> wm , . rr /. (67), (68)/ · wq = f1 ( w p ) f 2 (| p - q n |) rr rr · d = d + wq I (q n )
C (q n ) > C (q

r qm: r r m q n -1 + r : |m|

r

r rd d= W

·

W = W + wq

f 1 ( wC ) f 2 (t ) :

f1 ( w p ) = w

p

(67)
2

f 2 (t ) =
4.4

t

2

2 +t

(68)

. :


a AND b a b NOT a 1 - a

(69) (70)

a , b - . :

a OR b = NOT (( NOT a ) AND ( NOT b))
(69), (70) (71) :

(71) (72)

a OR b 1 - (1 - a ) (1 - b)

(72) «» . , a = b = 0.5 , a OR b = 0.75 , . , , . YUV, , , , , HSV. (66) Y,U,V : r r Y (d) = ( 0.299 0.587 0.114) T d r r (73) U (d ) = ( -0.147 - 0.289 0.436) T d r r T V (d ) = ( 0.615 - 0.515 - 0.1) d v :

r

r v = (U ,V )

(74)

: R1 : , (« »). R2 : , .

, ( )

Is(Y ) rr Ce( v 1 , v 2 )

max(0,1 - Y ) rr max(0,1 - | v 1 - v 2 |)

Ie(Y1 , Y2 )

Is(| Y1 - Y2 |)

: = 1 3 noise , noise - ( noise 0 255 noise = 1 Â 3 ).


: = 1 10 noise . , , , . R1 R2 R1 R2 ( ):
c c

rr rr R2c (d1 , d 2 ) = ( NOT Is(Y1 )) AND ( NOT Is(Y2 )) AND Ce( v1 , v 2 ) AND Ie(Y1 , Y2 )
c

rr R1c (d1 , d 2 ) = Is(Y1 ) AND Is(Y2 )

(75) (76)

rr R2c De(d1 , d 2 ) - (
):

R1 R2 R1 ,

rr rr rr (77) De(d1 , d 2 ) = R1c (d1 , d 2 ) OR R2c (d1 , d 2 ) r1 r 2 d1 d 1 , r1 r 2 d 2 d 2 .
:

( d1 d 2 d 1 d 2 ) ( d 1 d 2 d1 d ), .

R3 :

r

1

r

1

r

2

r

2

r

2

r

1

r

1

r

2 2

d 2 ), . R3 , . R4 , ( ). , R3 , R4 . R3 R4 R3 R4 ( ):
c c

r

r r r r r r r R4 : ( d1 d12 ) ( d1 d 2 ) ( d12 d12 ) ( d 1 1 2

2 1

2

rr R3c (d1 , d12 , 1 r OR (De(d r R4c (d1 , 1

rr rr rr d12 , d 2 ) = (De(d1 , d12 ) AND De(d12 , d 2 )) OR 2 1 2 r1 r1 r 2 2 , d 2 ) AND De(d1 , d 2 )) 1 rrr rr rr d12 , d12 , d 2 ) = De(d1 , d12 ) OR De(d12 , d 2 ) 2 1 2 r 2 r1 r1 r 2 OR (De(d1 , d 2 ) OR De(d1 , d 2 )

(78)

(79)
c

rrrr R4c Le(d1 , d12 , d12 , d 2 ) - ( 1 2
):

R3 R4 R3 ,

rrrr rrrr rrrr (80) Le(d1 , d12 , d12 , d 2 ) = R3c (d1 , d12 , d12 , d 2 ) OR R4c (d1 , d12 , d12 , d 2 ) 1 2 1 2 1 2 r1 r2 d 1,i d1,i i - r1 r2 , d 2 , j d 2 , j j - . cij (43) i - j - :


rrrr cij = Le(d1,i , d12,i , d12 , j , d 2 , j ) 1 2
5.

(81)

. 1. «» , . «» (.(82)). 2. (37). 3. «» . , . 4. «» , . 5. (88). . , ( ) . . . , , , . , , . «» . '1,i , '1,i i - ,

2, j ,

2, j

- j - . C '1 ( x, y ) C 2 ( x, y ) -

. i - j - j - i ; C '1 ( x, y ) C 2 ( x, y ) :

bij = b( x, y )C '1 ( x, y )C 2 ( x, y )dl b( x, y ) = x cos '1,i + y sin '1,i - '1,
i

(82)

j - , :

x cos

2, j

+ y sin

2, j

-

2, j

=0

(83)

«» :


b
:

2

ij

= b 2 ( x, y )C '1 ( x, y )C 2 ( x, y )dl

(84)



2

ij

=b

2

ij

- (bij )
ij

2

(85)

j - , b b
2 j
2


2

:

b
2

2

j

= min b i
2

ij

(86)
2 2 2

m b j , j , j j - :

m 2 = median b
j

2

j

2 = median j
ws = m 2 + 3

2

(87)
j

ws :
2

(88)

6.

, : - : , ; : · ( 2.2, (15)) · () ( 2.4, 2) · ( 2.4, 3) · ( 4.3, 4.4, 5) · ( 4.1, 4.2, 4), · ( 4) .
7.

.8-11. , ­ - . , , .


. 8. «». : - , - , - . : - ,

. 9. «». , . 8


. 10. «». , . 8

. 11. «». , . 8

Intel Pentium III, 800 .


t , t

a lg

: t
proj

radon



, t
best



:

t
t
radon

a lg

=t

radon

+t

proj

+t

best

(89)
radon

= O( N log N ) , N . 1024 â 1024 t radon
: t 1 . t
proj

.



N

= O ( N 1 N 2 ) , . N1 , N 2 10-15 , 1, t proj t
proj

1



N

2

:

100 .
1. .

, , % , % t

-180 -70 50

180 70 200

best

.
best

(3-5), t

= O( N ) .

1024 â 1024 t
8.

best

.

, . . , . , , [34, 36, 37]. . -, , . -, , . -, , . «» . , 3D . , , «» .


9.

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