Lecture 3 Camera Models 2 & Camera Calibration. Professor Silvio Savarese Computational Vision and Geometry Lab

Transcription

1 Lecture Cer Models Cer Clbrton rofessor Slvo Svrese Coputtonl Vson nd Geoetry Lb Slvo Svrese Lecture - Jn 7 th, 8

2 Lecture Cer Models Cer Clbrton Recp of cer odels Cer clbrton proble Cer clbrton wth rdl dstorton Exple Redng: [F] Chpter Geoetrc Cer Clbrton [HZ] Chpter 7 Coputton of Cer Mtrx Soe sldes n ths lecture re courtesy to rofs. J. once, F-F L Slvo Svrese Lecture - Jn 7 th, 8

3 rojectve cer j w R, f O w k w ' M w K " R! " $! W % " W W W $ % $ % 4 w4 M E ' E ( w, w ) w w

4 Exercse! f R, j w k w O w w à M K! " R $ ( w, w ) w w Suppose we hve no rotton or trnslton Zero skew, squre pxels, no dstorton, no off-set ' E w! " x w y w z w $ %

5 Exercse! R, j w f k w O w w à M K! " R ' E $ ( w w, w w ) K! " I $! " ( f x w z w, f y w z w ) f f $ %! w " x w y w z w $ %

6 Cnoncl rojectve rnsforton ' x y z! " $ %! " $ % x y z! " $ % x z y z! " $ % ' ' M M H 4 Â Â

7 rojectve cer p q r f O Q R

8 Wek perspectve projecton When the reltve scene depth s sll copred to ts dstnce fro the cer f p R R_ q r Q Q_ O zo _ π

9 Wek perspectve projecton f zo p R_ q r Q Q_ O R! " $ x' f ' z x y' f ' z y! " $ x' f ' z x y' f ' z y _ π Mgnfcton

10 Wek perspectve projecton f zo p R_ q r Q Q_ O π R _ rojectve (perspectve) Wek perspectve M K [ R ] A v b à M A b

11 ! " $ % W ), ( w w E M w b A M gnfcton M w v b A M W W W W ), ( w w w w E erspectve Wek perspectve W W! " $ %

12 Orthogrphc (ffne) projecton Dstnce fro center of projecton to ge plne s nfnte! " $ x' f ' z x y' f ' z y! " $ x' x y' y

13 ros nd Cons of hese Models Wek perspectve results n uch spler th. Accurte when object s sll nd dstnt. Most useful for recognton. nhole perspectve s uch ore ccurte for odelng the D-to-D ppng. Used n structure fro oton or SLAM.

14 One-pont perspectve Mscco, rnty, Snt Mr Novell, Florence, 45-8 Credt slde S. Lzebnk

15 Wek perspectve projecton he Kngx Eperor's Southern Inspecton our (69-698)By Wng Hu

16 Wek perspectve projecton he Kngx Eperor's Southern Inspecton our (69-698)By Wng Hu

17 Lecture Cer Clbrton Recp of cer odels Cer clbrton proble Cer clbrton wth rdl dstorton Exple Redng: [F] Chpter Geoetrc Cer Clbrton [HZ] Chpter 7 Coputton of Cer Mtrx Soe sldes n ths lecture re courtesy to rofs. J. once, F-F L Slvo Svrese Lecture - 8-Jn-8

18 Why s ths portnt? Estte cer preters such pose or focl length fro ges!?

19 rojectve cer - v u cot K o o sn q b q R r r r z y x t t t 4 M w [ ] w K R Internl preters Externl preters

20 Gol of clbrton K[ R ] w M w Internl preters Externl preters Estte ntrnsc nd extrnsc preters fro or ultple ges Chnge notton: w p

21 Clbrton roble Clbrton rg j C n wth known postons n [O w, w,j w,k w ]

22 Clbrton roble Clbrton rg ge p j C n wth known postons n [O w, w,j w,k w ] p, p n known postons n the ge Gol: copute ntrnsc nd extrnsc preters

23 Clbrton roble Clbrton rg ge p j C p j C How ny correspondences do we need? M hs unknowns We need equtons 6 correspondences would do t

24 Clbrton roble Clbrton rg ge p j C p j C In prctce, usng ore thn 6 correspondences enbles ore robust results

25 Clbrton roble j C Clbrton rg p j C p p ge M p u v! " $ % n pxels [Eq. ] M

26 Clbrton roble u ) ( v ) ( u v v u ) ( v - u ( ) [Eq. ] [Eqs. ]

27 Clbrton roble u ( ) - v ( ) - u v ( ) - ( ) - [Eqs. ] u v n n ( n ) - n ( n ) - n

28 Block Mtrx Multplcton B B B B B A A A A A Wht s AB? B A B A B A B A B A B A B A B A AB

29 - u ( ) + - v ( ) + - un( n ) + n Clbrton roble - vn ( n ) + n known unknown [Eq. 4] Hoogenous lner syste def " $ $ $ $ $ $ $! n n u v u n n v n n x4 % ' ' ' ' ' ' ' n x def æ ç ç ç è 4x ö ø x

30 Hoogeneous M x N Lner Systes Mnuber of equtons n Nnuber of unknown M N Rectngulr syste (M>N) s lwys soluton o fnd non-zero soluton Mnze under the constrnt

31 Clbrton roble How do we solve ths hoogenous lner syste? V SVD decoposton!

32 Clbrton roble SVD decoposton of U D V n Lst colun of V gves Why? See pg 59 of HZ def æ ç ç ç è ö ø M

33 Extrctng cer preters M ρ

34 Extrctng cer preters A A [ ] K R ± r b b b b Estted vlues ( ) r u o ) ( v o r ( ) ( ) cos q Intrnsc b - v u cot K o o sn q b q r M Box See [F], Sec...

35 Extrctng cer preters A [ ] K R b b b b Estted vlues Intrnsc q r sn q r b sn A b r M - v u cot K o o sn q b q Box See [F], Sec...

36 heore (Fugers, 99)

37 Extrctng cer preters A [ ] K R b b b b Estted vlues A b r M - v u cot K o o sn q b q Box See [F], Sec... Extrnsc ( ) r r ± r r r b K - r

38 Degenerte cses s cnnot le on the se plne! onts cnnot le on the ntersecton curve of two qudrc surfces [F] secton.

39 Lecture Cer Clbrton Recp of projectve cers Cer clbrton proble Cer clbrton wth rdl dstorton Exple Redng: [F] Chpter Geoetrc Cer Clbrton [HZ] Chpter 7 Coputton of Cer Mtrx Soe sldes n ths lecture re courtesy to rofs. J. once, F-F L Slvo Svrese Lecture - 8-Jn-8

40 Rdl Dstorton Ige gnfcton (n)decreses wth dstnce fro the optcl xs Cused by perfect lenses Devtons re ost notceble for rys tht pss through the edge of the lens No dstorton n cushon Brrel

41 Rdl Dstorton p v u M l l d v v c u b v u d + + u o odel rdl behvor ± å p p κ p d λ olynol functon Dstorton coeffcent [Eq. 5] [Eq. 6] S λ Ige gnfcton decreses wth dstnce fro the optcl center How do we odel tht?

42 v u p Q q q q q q q q p v u M l l Q u q q v q q! " $ Is ths lner syste of equtons? Rdl Dstorton No! why? [Eqs.7]

43 Generl Clbrton roble u v q q q q n X f (Q) [Eq.8] preters esureents f( ) s the nonlner ppng -Newton Method -Levenberg-Mrqurdt Algorth Itertve, strts fro ntl soluton My be slow f ntl soluton fr fro rel soluton Estted soluton y be functon of the ntl soluton (becuse of locl n) Newton requres the coputton of J, H Levenberg-Mrqurdt doesn t requre the coputton of H

44 Generl Clbrton roble u v q q q q n X f (Q) [Eq.8] preters esureents f( ) s the nonlner ppng A possble lgorth. Solve lner prt of the syste to fnd pproxted soluton. Use ths soluton s ntl condton for the full syste. Solve full syste usng Newton or L.M.

45 ypcl ssuptons: - zero-skew, squre pxel - u o, v o known center of the ge Generl Clbrton roble esureents preters f( ) s the nonlner ppng q q q q v u X f (Q) [Eq.8] n

46 Cn we estte nd nd gnore the rdl dstorton? Rdl Dstorton v u p l d v u Hnt: slope v u q q q q

47 Esttng nd Rdl Dstorton v u p l ) ( ) ( - u v ) ( ) ( - u v ) ( ) ( - n n n n u v L n n! " $ % s [87] v u ) ( ) ( ) ( ) ( Get nd by SVD [Eq.9] [Eq.] [Eq.] L def v u v u!! v n n u n n " $ $ $ $ $ % ' ' ' ' '

48 Once tht nd re estted Rdl Dstorton v u p l s non lner functon of l here re soe degenerte confgurtons for whch nd cnnot be coputed,,

49 Lecture Cer Clbrton Recp of projectve cers Cer clbrton proble Cer clbrton wth rdl dstorton Exple Redng: [F] Chpter Geoetrc Cer Clbrton [HZ] Chpter 7 Coputton of Cer Mtrx Soe sldes n ths lecture re courtesy to rofs. J. once, F-F L Slvo Svrese Lecture - 8-Jn-8

50 Clbrton rocedure Cer Clbrton oolbox for Mtlb J. Bouguet [998-]

51 Clbrton rocedure

52 Clbrton rocedure

53 Clbrton rocedure

54 Clbrton rocedure

55 Clbrton rocedure

56 Clbrton rocedure

57 Clbrton rocedure

58 Next lecture Sngle vew reconstructon

59 Egenvlues nd Egenvectors Egendecoposton A - l l SL S S S. l N - Egenvectors of A re coluns of S [ ] S v v N

60 Sngulr Vlue decoposton A U S V - S U, V orthogonl trx s s. s N s l s sngulr vlue l egenvlue of A t A