Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://imaging.cs.msu.ru/~yurin/FundamentalMatrix.pdf Äàòà èçìåíåíèÿ: Sun Oct 19 06:02:34 2008 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 19:41:38 2012 Êîäèðîâêà:

FundamentalMatrix.doc

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10/19/2008



.

(pinhole camera).
, .
(xp , yp , z p )





j


f

(xp , y p )

k

s

f

p

~~ u fp , v

fp

z
k

p

j
c

c

t

f

i
l
f

i

c

f

. 1. . t ,

f = 1..F , , r
r p = 1..P s p ,
f

().

~~ u fp , v fp - () . ,
(. . 2) [1,2]:

u

fp

=g

i f (s p - t f )
f

k f (s p - t f )

,v

fp

=g

j f (s p - t f )
f

k f (s p - t f )

, (1)

lf g f = ~ e , uN
f

~ u fp u fp = ~ e , uf N

~ v fp v fp = ~ e , uN
f

rrr i f , j f , k f - , , r rr f - (), k f , i f , j f ~e ; l f - . u f - ; N - . ~~ gf, u fp , v fp [ - 1 2 ,1 2] l f , u fp , v fp .
(1) :

fp u fp = g f i f (s p - t f ) fp v fp = g f j f (s p - t f ) fp = k f (s p - t f )
, ~e ~e ~e ~e x y u f v f aspect ratio: = v f / u f . : , (2)


FundamentalMatrix.doc

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gf Kf = 0 0 Rf =


0 g 0

f

(i

0 0 1

f

j
fp
T

f

k
fp

f

)

,

(3)

T

r u

fp

= =

(u

v

1)

T

S (s p 1)
p

(4)

u

r

fp

3

S

p

4 .

gf K f Rf = 0 0

0

g
0 =

f

(g

0 0 1 f

(i
f

f

j

f

k
f

f

)

T

gf = 0 0 f

0 g 0

f

i

g f j

k

)
(

0 i Tf g f i Tf 0 jTf = g f jTf = 1 k Tf k Tf ,

T

(2)

fp u fp = K f R f (s p - t f ) = K f R f - R f t f S
, :

r

)

p

,

(5)

Pf = R f - R f t

(

M f = K f Pf = K f R f - K f R f t Qf =KfRf r q f = -K f R f t


(

f

)

f

) = (Q

f

r q

f

)

, Pf , Q f

3â4

3â4 3â3

f

, Q f r , q f 3

,

(6)

fp u fp = M f S
(1) .

r

p

(7)

r u

fp

~ M f Sp,

(8)

« ». , (1) , , « » - « » (. .1). u

r

fp

3

( ) , . (7), , , «» . (7) , , ( ) ().


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[3,4] (7) (6)

fp u

r

fp

r = Qf q

(

f

)

r r S p = Q f sp + q

f

(9)

, :

1 p u
Q

r

2 p u
f

r

1p 2p

r r = Q1 s p + q1 r r = Q2 sp + q
3â3

(10)
2

=KfR

f

, Q f

(3), :

r r r - - Q1 1 (1 p u1 p - q 1 ) = Q 1 1 Q 1 s p r r, r Q -1 ( 2 p u 2 p - q 2 ) = Q -1 Q 2 s p 2 2
, , :

(11)

r - Q 2 Q 1 1 (1 p u


1p

r r - q 1 ) = ( 2 p u

2p

r - q 2 ),

(12)

H p n
(12) :

12

21 2p

- = Q 2 Q1 1 r r = q 2 - H 12 q1 , r = - H 12 u1 p

(13)

2 p u

r

2p

r + 1 p n

2p

= p 21 ,

(14)

p - ,



fp

. : , , ­ , .




2 u + 1n = p ,
:

r

r

r

(15)

2 u1 + 1n1 = p1 2 u 2 + 1n2 = p 2 u + n = p 13 3 23
:

,

p1 p 2 = 2 u1 u2

n1 n2 p n - p 2 n1 = 12 n1 u1 n2 - u 2 n1 n2

u1

p1 p 2 u1 p 2 - u 2 p1 , = n1 u1n2 - u 2 n1 n2

, 2 =

u2 u1 u2

, , :


FundamentalMatrix.doc

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p1n2 - p2 n1 u p - u 2 p1 u3 + 1 2 n3 - p3 = 0 . u1n2 - u 2 n1 u1n2 - u 2 n1
, , :

( p1 n2 - p 2 n1 )u 3 + (u1 p 2 - u 2 p1 )n3 - p3 (u1 n 2 - u 2 n1 ) = 0 p1 n 2 u 3 - p 2 n1u 3 + p 2 n3u1 - p1 n3u p1 eijk pi n j u k = p1 p1 n1 n2 n3
23

- p3 n 2 u1 + p3 n1u 2 =
.

p1 n 2 u 3 + p 2 n3 u1 + p3 n1u 2 - p1 n3u 2 - p 2 n1u 3 - p3 n2 u1 =


ijk

u1 rrr rrr u 2 = (p [n â u]) = (p, n, u) = 0 u3

e

ijk

- 3-.





2 u + 1n = p ,
rr , u , n : r r rrrrr ( 2 u + 1n ) [u â n] = p [u â n] .


r

r

r

(15)

rrr rrr rrr p [u â n] = (p, n, u) = det (p n u ) = 0 .

(16)


, , rr , .. u , n . (15) (14) (13) :

r 0u

2p

ân

2p

= -u

2p

r â H12 u

1p

= -u

2p

-r â Q 2 Q1 1u1 p .

(17)

(15) (14) :

r r p 21 [u

2p

r â n2p ] = 0 .

(18)

(13) :

r r - p 21 [u
(19) :

2p

r â H 12 u1 p ] = 0 ,

(19)

-

e ( p ) (u ) (
ijk 21 i ijkl 2p j

H

12 kl

) (u1 p )l = (u 2 p )
2p j

=

(u )
jl

F jl (u1

jl

j

pl

)

-

e (p ) (
ijk 21 i ik

H

12 kl

) (u1 p )l


,

(20)

=0

,

r r u 2 p Fu

1p

=0,

(21)


Comment: !!!

FundamentalMatrix.doc
F ­ .

5 F jl = -

10/19/2008 H

e (p ) (
ijk

21 i

12 kl

)

,

ik

(22)

(a.14) , (a.15) , , (22) :

r [p 21 ]â = -

e (p )
ijk l 12

21 i

r F = [p 21 ]â H

,

(23)

, :

0 r F = [p 21 ]â H12 ( p21 )3 - (p ) 21 2
det F = 0


- (p

21 3

)

( p21 )1

0

- (p

( p21 )2 21 )1
0

H

12

,

(. (.15)) (23)
-1 1

H12 = Q 2 Q

Qf =KfRf r r p 21 = q 2 - H12 q1 r q f = -K f R f t
c. (1)-(3).

f


(21)

r r u 2 p Fu

1p

= 0,

det F = 0

(24)

(23). , , , (24). 9 7 . (24) , 1 2 ( ). r u10 = ( x0 , y 0 ,1) , ,

r = l , , , rT r u 2 p l = 0 , - r Fu
1p

(). .


FundamentalMatrix.doc

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FundamentalMatrix.doc
Civita_symbol .

7

10/19/2008

. .
«Levi-Civita symbol» . http://en.wikipedia.org/wiki/Levi k [5] i1i2 ...ik ,

k â k â ... â k , -1, 0, +1. e 14243
k

i j : k , i j = 1,2,..., k .
: 1 :

e12...k = 1 , .

, ­ .

ei e1 1
e11
21

2 : eij = e 3 : e .
ijk

e12 0 1 e22 - 1 0
213

: e123 = 1 , e

= -1 , e

231

= 1 , e321 = -1 , e

312

= 1 , e132 = -1 .

k k!, k k!. , k! kk. 3- , : , , , {123}, -1 . 3, . . , a b a ijk b



, , .
ki


ki

aijk bki . -

, . , . .

1. ( )
, [6] . , ­ , . 3. « 1,...,n , - 1,2 : 1,2 2,1. 1,...,n i1,i2,...,in. , in i1,i2,...,in, . n=4 2,4,3,1 2 3 , 4 ­ . . i1,i2,...,in N(i1,i2,...,in). i1,i2,...,in , N(i1,i2,...,in) - , . :


FundamentalMatrix.doc
a11 L a1
n

8

10/19/2008

det( A) = L L L = a n1 L a nn
»

( i1Kin )



( -1)

N ( i1Kin )

a1i1 a

21i

2

K a nin ,

(a.1)

n ! , :

rr r det( A) = det (a1 a 2 K a


n

)=
i1Ki
n

e

i1Ki

n

a1i1 a

21i

2

K a nin ,

(a.2)

r r a k - k- - A, a kin - in- .

i1,i2,...,in, i1,i2,...,in ik=1...n, k=1...n. - . , 33

rrr A= ab c B= r a r b r c
T T T

(

) (

, det A =


i =1 j =1 k =1 T

3

3

3

eijk ai b j ck = eijk ai b j c

def

k

rrr = abc

)

T

=A

, det B =


i =1 j =1 k =1

3

3

3

eijk ai b j c k = eijk ai b j c

def

,
k

(a.2)

2. :
i- :

[ a â b]i = eijk a j bk

3. :
r rr rrr (ab c ) = (a [b â c ]) = eijk a i b j c k .

4. () .
( ) . , . ( ) [5].

eijk e ,

(a.3)

, :

eijk e

= c

i j k



i j k



i j k

,

(a.4)


FundamentalMatrix.doc

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, , , , . , . , , -1,0,1. .

eijk ei



=


i =1

3

eijk ei



=c


i =1

3



ii ji ki



i j k



i j k

1 = c0 0

i j k



i j k

= c( j

k

- j

k

)

,

(a.5)

:

eijk eij = c ( jj


k

- j

kj

) = c(3
kk

k

-

k

)

= 2c k ,

(a.6)

eijk e


ijk

= 2c

= 6c ,

(a.7)

eijk eijk =


ijk

2 eijk = { } = 3! = 6 ,

(a.8)

=1 :

eijk e

=

i j k



i j k



i j k

eijk ei



= ( j k - j
k

k

)
,

(a.9

eijk eij = 2

eijk eijk = 3! = 6

( :

r rr (a â [b â c ]) i = eijk a j ek b c = eijk e


k

a j b c





=

ekij e

k

a j b c

( .9 )



=

(

i

j - i

j

)a

j

b c =
(a.10)

rrr rr r rr rr = (a j bi c j - a j b j ci ) = bi (a c ) - ci ( a b ) = b (a c ) - c ( a b ) r r r rrr rr r a â [b â c ] = b ( a c ) - c ( a b )

(

)

i

, ..

,

, ,

5.



FundamentalMatrix.doc

10
eijk a j ,

10/19/2008
(a.11)

aj ­ . , .. . -:

eijk a j = (eijk a j ) ik = -(ekji a j ) e123 a 2 = (e123 a 2 )13 = a e231 a3 = (e231 a3 ) e312 a1 = (e312 a1 ) 0 (eijk a j ) = a3 - a
:
21 2 3

ki

=a -a 0

, (a.12)

32

= a1
3

2

a1

a2 - a1 0

= -(eikj ai ) = a1
2 3

eijk a k = (eijk a k ) ij = -(e jik a k ) e123 a3 = (e123 a 2 )12 = a e231 a1 = (e231 a1 ) 0 (eijk a k ) = - a a 2
23 3

ji

eijk ai = (eijk ai )
,

jk 23

kj

e123 a1 = (e123 a1 )

= a1
2

e231 a 2 = (e231 a 2 ) 31 = a e312 a3 = (e312 a3 )12 = a 0 (eijk ai ) = - a a 2 a
3 3

e312 a 2 = (e312 a 2 ) 31 = a a
3 3

(a.13)

0 - a1

- a2 a1 = -(eikj a k ) 0

0 - a1

- a2 a1 = -(e jik a i ) 0

­ , :

0 r def [a]â = a z - a

-a 0
y

z

a

x

ay 0 def - a x = a3 0 - a

-a 0
2

3

a1

a2 - a1 0

,

(a.14)

(a.13) :

r def [a]â = (eijk a j ) r [a]â = -(eijk ai ) , r [a]â = -(eijk a k )

r r det([a]â ) = 0 , rank ([a]â ) 2 , r

(a.15)

[a]â , a , , , ( , , 1 0 ­ ).

r

.
1. 2. . . , . . . . // 2004, .30, 5, . 48-68. . . , .. . , , . ­ . . 12- '2002 ­ . 123-129. , 2002. http://www.graphicon.ru/2002/pdf/Yanova_Re.pdf G. Csurka, C. Zeller, Z. Zhang, and O. Faugeras. Characterizing the uncertainty of the fundamental matrix. Technical Report 2560, I.N.R.I.A., France, 1995. http://citeseer.ist.psu.edu/article/csurka95characterizing.html

3.


FundamentalMatrix.doc
4.

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10/19/2008

5. 6. 7.

8.

9.

Gabriella Csurka and Cyril Zeller and Zhengyou Zhang and Olivier D. Faugeras. Characterizing the Uncertainty of the Fundamental Matrix, Computer Vision and Image Understanding: CVIU, 1997 -Vol 68, No 1, P 18-36, http://citeseer.ist.psu.edu/article/csurka95characterizing.html .. .. : . . 10 . . II . -7- .. ­.:. . . .-. . 1988. 512 . ­ISBN 5-02-014420-7 (. II). § 6. ... . -- . -6- ., . -.:, 1987. -320. Conrad I. Poelman, Takeo Kanade. A Paraperspective Factorization Method for Shape and Motion Recovery: //Technical Report CMU-CS-93-219 / School of Computer Science, Carnegie Mellon http://www.ri.cmu.edu/pubs/pub_1189.html, University. -- 11 December 1993. http://www.ri.cmu.edu/people/person_136_pubs.html Carlo Tomasi, Takeo Kanade. Shape and Motion from Image Streams: a Factorization Method, Part 3, Detection and Tracking of Point Features //Technical Report CMU-CS-91-132 / School of Computer Science, Carnegie Mellon University. -- April 1991. http://www.ri.cmu.edu/pubs/pub_2543.html Anton Shokurov, Andrey Khropov, Denis Ivanov. Feature Tracking in Images and Video. ­ . . 13- '2003 ­ . 177-179. , 2003.

10. Joao Paulo Salgado Arriscado Costeira. A multi-body Factorization method for motion analysis: //Tese para obtencao do grau de doutor em Engenharia Electrotecnica e de Computadores. /Universitade Technica de Lisboa Instituto Superior Rechnico. Lisboa, Maio de 1995. http://omni.isr.ist.utl.pt/~jpc/pubs.html
11. . . . ­ .: , 2001.