Lecture 8: Camera Calibration

Transcription

1 Lecture 8: Cer Clbrton rofessor Fe-Fe L Stnford Vson Lb Fe-Fe L 9-Oct-

2 Wht we wll lern tody? Revew cer preters Affne cer odel (roble Set (Q4)) Cer clbrton Vnshng ponts nd lnes (roble Set (Q)) Redng: [F] Chpter [HZ] Chpter 7, 8.6 Fe-Fe L 9-Oct-

3 Wht we wll lern tody? Revew cer preters Affne cer odel Cer clbrton Vnshng ponts nd lnes Redng: [F] Chpter [HZ] Chpter 7, 8.6 Fe-Fe L 9-Oct-

4 f rojectve cer O c f focl length Fe-Fe L 4 9-Oct-

5 f rojectve cer O c y y c x c f focl length u o, v o offset C[u o, v o ] x Fe-Fe L 5 9-Oct-

6 f rojectve cer O c Unts: k,l [pxel/] f [] Non-squre pxels α, β [pxel] f focl length u o, v o offset α, β non-squre pxels Fe-Fe L 6 9-Oct-

7 f rojectve cer c O c ' α s β u v o o K hs 5 degrees of freedo! x y z f focl length u o, v o offset α, β non-squre pxels θ skew ngle Fe-Fe L 7 9-Oct-

8 f rojectve cer c O c α α cotθ β snθ u v o o K hs 5 degrees of freedo! x y z f focl length u o, v o offset α, β non-squre pxels θ skew ngle Fe-Fe L 8 9-Oct-

9 f rojectve cer R,T c j w k w O w O c w T R T 4 ~ RO c 4 w f focl length u o, v o offset α, β non-squre pxels θ skew ngle R,T rotton, trnslton Fe-Fe L 9 9-Oct-

10 f rojectve cer R,T j w k w O w O c w M w K[ R T] w Internl (ntrnsc) preters Externl (extrnsc) preters f focl length u o, v o offset, non-squre pxels θ skew ngle R,T rotton, trnslton α β Fe-Fe L 9-Oct-

11 rojectve cer K[ R T] w M w Internl (ntrnsc) preters Externl (extrnsc) preters Fe-Fe L 9-Oct-

12 Fe-Fe L 9-Oct- rojectve cer M w [ ] w T K R v u cot K o o snθ β θ α α T T T R r r r z y x t t t T 4

13 Fe-Fe L 9-Oct- Gol of clbrton M w [ ] w T K R v u cot K o o snθ β θ α α T T T R r r r z y x t t t T 4 Estte ntrnsc nd extrnsc preters fro or ultple ges

14 Wht we wll lern tody? Revew cer preters Affne cer odel (roble Set (Q4)) Cer clbrton Vnshng ponts nd lnes Redng: [F] Chpter [HZ] Chpter 7, 8.6 Fe-Fe L 4 9-Oct-

15 Wek perspectve projecton x' x y' y where f z ' gnfcton Reltve scene depth s sll copred to ts dstnce fro the cer Fe-Fe L 5 9-Oct-

16 Orthogrphc (ffne) projecton x' y' x y Dstnce fro center of projecton to ge plne s nfnte Fe-Fe L 6 9-Oct-

17 Fe-Fe L 9-Oct- 7 Affne cers [ ] T R K ' s K y x α α T R K M Affne cse rllel projecton trx T R K M y x s K o y o x α α rojectve cse Copred to

18 Fe-Fe L 9-Oct- 8 Reeber. rojectvtes: y x H y x b v t A y' x' p Affntes: y x H y x t A y' x'

19 Fe-Fe L 9-Oct- 9 [ ] T R K ' y x K α α T R K M b A 4ffne] [4 ffne] [ b b M + + ' M b b Z Y X y x Euc b A [ ] b A M M Euc We cn obtn ore copct forulton thn: Affne cers

20 Affne cers To recp: M cer trx ' u v A + b M ; M Ths notton s useful when we ll dscuss ffne structure fro oton [non-hoogeneous ge coordntes] [ A b] Fe-Fe L 9-Oct-

21 Affne cers Wek perspectve uch spler th. Accurte when object s sll nd dstnt. Most useful for recognton. nhole perspectve uch ore ccurte for scenes. Used n structure fro oton. Fe-Fe L 9-Oct-

22 Wek perspectve projecton -exples The Kngx Eperor's Southern Inspecton Tour (69-698) By Wng Hu You tube vdeo clck here Fe-Fe L 9-Oct-

23 Wek perspectve projecton -exples Qngng Festvl by the Rversde Zhng Zedun ~9 AD Fe-Fe L 9-Oct-

24 Wht we wll lern tody? Revew cer preters Affne cer odel Cer clbrton Vnshng ponts nd lnes Redng: [F] Chpter [HZ] Chpter 7, 8.6 Fe-Fe L 4 9-Oct-

25 Clbrton roble Clbrton rg 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 Fe-Fe L 5 9-Oct-

26 Reeber the dgtl Mchelngelo project? Fe-Fe L 6 9-Oct-

27 Clbrton roble Clbrton rg j C How ny correspondences do we need? Mhs unknown We need equtons 6 correspondences would do t Fe-Fe L 7 9-Oct-

28 Clbrton roble Clbrton rg ge j C In prctce: user y need to look t the ge nd select the n>6 correspondences Fe-Fe L 8 9-Oct-

29 Fe-Fe L 9-Oct- 9 Clbrton roble j C M p v u p M n pxels

30 Fe-Fe L 9-Oct- Clbrton roble u ) ( v ) ( u v v u ) ( v ) ( u

31 Fe-Fe L 9-Oct- Clbrton roble ) ( v ) ( u ) ( v ) ( u ) ( n n n v ) ( n n n u

32 Fe-Fe L 9-Oct- 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

33 Clbrton roble u ( ) + v ( ) + known unknown u v n n ( n ) + n ( n ) + n Hoogenous lner syste x4 n x def 4x T T T x Fe-Fe L 9-Oct-

34 Hoogeneous M x N Lner Systes Mnuber of equtons Nnuber of unknown A x Rectngulr syste (M>N) s lwys soluton To fnd non-zero soluton Mnze Ax under the constrnt x Fe-Fe L 4 9-Oct-

35 Clbrton roble How do we solve ths hoogenous lner syste? Sngulr Vlue Decoposton (SVD) Fe-Fe L 5 9-Oct-

36 Clbrton roble Copute SVD decoposton of U D V T n Lst colun of V gves Why? See pge 59 of Hrtley & Zssern M M p Fe-Fe L 6 9-Oct-

37 Fe-Fe L 9-Oct- 7 Extrctng cer preters A T T T A [ ] T K R ± ρ b b b b Estted vlues ) ( u o ρ ) ( v o ρ ( ) ( ) cos θ Intrnsc b v u cot K o o snθ β θ α α ρ

38 Fe-Fe L 9-Oct- 8 Theore (Fugers, 99) [ ] [ ] ] [ b A K T K R T R K M A y x c c s K β α l f k; f β α

39 Fe-Fe L 9-Oct- 9 Extrctng cer preters A T T T A [ ] T K R b b b b Estted vlues Intrnsc θ ρ α sn θ ρ β sn b f ρ

40 Fe-Fe L 9-Oct- 4 Extrctng cer preters Extrnsc ( ) r r ± r r r b K T ρ A T T T A [ ] T K R b b b b Estted vlues b ρ

41 Clbrton Deo Cer Clbrton Toolbox for Mtlb J. Bouguet [998-] Fe-Fe L 4 9-Oct-

42 Clbrton Deo Fe-Fe L 4 9-Oct-

43 Clbrton Deo Fe-Fe L 4 9-Oct-

44 Clbrton Deo Fe-Fe L 44 9-Oct-

45 Clbrton Deo Fe-Fe L 45 9-Oct-

46 Clbrton Deo Fe-Fe L 46 9-Oct-

47 Clbrton Deo Fe-Fe L 47 9-Oct-

48 Clbrton Deo Fe-Fe L 48 9-Oct-

49 Wht we wll lern tody? Revew cer preters Affne cer odel Cer clbrton Vnshng ponts nd lnes (roble Set (Q)) Redng: [F] Chpter [HZ] Chpter 7, 8.6 Fe-Fe L 49 9-Oct-

50 ropertes of rojecton onts project to ponts Lnes project to lnes Fe-Fe L 5 9-Oct-

51 ropertes of rojecton Angles re not preserved rllel lnes eet Vnshng pont Fe-Fe L 5 9-Oct-

52 Fe-Fe L 9-Oct- 5 Lnes n D plne c by x + + -c/b -/b c b l If x [ x, x ] T l c b x x T l x y

53 Lnes n D plne Intersectng lnes x l l l roof l l l l l l ( l l ) l ( l l ) l x y x l x l x l x s the ntersectng pont x Fe-Fe L 5 9-Oct-

54 Fe-Fe L 9-Oct- 54 onts t nfnty (del ponts) x, x x x x c b l c b l b ) c (c l l Let s ntersect two prllel lnes: Agree wth the generl de of two lnes ntersectng t nfnty l l x x x

55 Fe-Fe L 9-Oct- 55 Lnes t nfnty l Set of del ponts les on lne clled the lne t nfnty How does t look lke? l l T x x Indeed:

56 Fe-Fe L 9-Oct- 56 rojectve projectons of lnes t nfnty (D) l H l T b v t A H? l H T b t t b v t A y x T s t lne t nfnty? no!? l H T A T T T T A t A t A

57 rojectve projectons of lnes t nfnty (D) horzon l hor T H l Are these two lnes prllel or not? Recognton helps reconstructon! Huns hve lernt ths - Recognze the horzon lne -Mesure f the lnes eet t the horzon -f yes, these lnes re // Fe-Fe L 57 9-Oct-

58 Vnshng ponts ( del ponts n D) d Vnshng ponts ponts where prllel lnes ntersect n D v C ddrecton of the lne M K[ R T ] Ige of vnshng pont v K d Fe-Fe L 58 9-Oct-

59 Horzon Sets of prllel lnes on the se plne led to collner vnshng ponts [The lne s clled the horzonfor tht plne] horzon Fe-Fe L 59 9-Oct-

60 Horzon n l horz C T n K l horz Fe-Fe L 6 9-Oct-

61 Applcton These trnsfortons re used n sngle vew etrology Crns & Zssern, 99 Fe-Fe L 6 9-Oct-

62 Applcton these trnsfortons re used n sngle vew etrology Crns & Zssern, 99 Fe-Fe L 6 9-Oct-

63 Applcton these trnsfortons re used n sngle vew etrology Crns & Zssern, 99 L Trnt'(46) Frenze, Snt Mr Novell; by Mscco (4-48) Fe-Fe L 6 9-Oct-

64 Fe-Fe L 64 9-Oct-

65 Applcton these trnsfortons re used n sngle vew etrology Hoe et l, 5 Fe-Fe L 65 9-Oct-

66 Applcton these trnsfortons re used n sngle vew etrology Sxen, Sun, Ng, 5 A softwre: MkeD Convert your ge nto d odel Fe-Fe L 66 9-Oct-

67 Wht we hve lerned tody Revew cer preters Affne cer odel (roble Set (Q4)) Cer clbrton Vnshng ponts nd lnes (roble Set (Q)) Redng: [F] Chpter [HZ] Chpter 7, 8.6 Fe-Fe L 67 9-Oct-

68 Suppleentry Mterls Fe-Fe L 68 9-Oct-

69 Degenercy nd dstorton n rel-world cer clbrton Fe-Fe L 69 9-Oct-

70 Degenerte cses s cnnot le on the se plne! onts cnnot le on the ntersecton curve of two qudrc surfces Fe-Fe L 7 9-Oct-

71 Cused by perfect lenses Rdl Dstorton Devtons re ost notceble for rys tht pss through the edge of the lens No dstorton n cushon Brrel Fe-Fe L 7 9-Oct-

72 Fe-Fe L 9-Oct- 7 Rdl Dstorton p v u M λ λ d v v u c v b u d + + u ± p p κ p d λ olynol functon Dstorton coeffcent To odel rdl behvor

73 Fe-Fe L 9-Oct- 7 Rdl Dstorton v u p Q q q q q q q q p v u M λ λ Q v u q q q q Non-lner syste of equtons

74 Generl Clbrton roble X f () f( ) s nonlner esureent preter -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 Newton requres the coputton of J, H Levenberg-Mrqurdt doesn t requre the coputton of H Fe-Fe L 74 9-Oct-

75 Generl Clbrton roble X f () f( ) s nonlner esureent preter 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 (ncludng dstorton) usng Newton or L.M. Fe-Fe L 75 9-Oct-

76 Generl Clbrton roble X f () f( ) s nonlner esureent preter Typcl ssuptons for coputng ntl condton : - zero-skew, squre pxel -u o, v o known center of the ge - no dstorton Just estte f nd R, T Fe-Fe L 76 9-Oct-

77 Fe-Fe L 9-Oct- 77 Ts s clbrton technque. Estte nd frst: v u p λ How to do tht? d v u Hnt: slope v u

78 Fe-Fe L 9-Oct- 78 Ts s clbrton technque. Estte nd frst: v u p λ ) ( ) ( u v ) ( ) ( u v ) ( ) ( n n n n u v Q n n v u ) ( ) ( ) ( ) (

79 Fe-Fe L 9-Oct- 79 Ts s clbrton technque. Once tht nd re estted, estte : v u p λ s non lner functon of λ There re soe degenerte confgurtons for whch nd cnnot be coputed