Lecture 8: Camera Calibra0on
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1 Lecture 8: Cer Clbron rofessor Fe- Fe L Stnford Vson Lb Lecture 8 -!
2 Wht we wll lern tody? Revew cer preters Affne cer odel (roble Set (Q4)) Cer clbron Vnshng ponts nd lnes (roble Set (Q)) Redng: [F] Chpter [HZ] Chpter 7, 8.6 Lecture 8 -!
3 Wht we wll lern tody? Revew cer preters Affne cer odel Cer clbron Vnshng ponts nd lnes Redng: [F] Chpter [HZ] Chpter 7, 8.6 Lecture 8 -!
4 f rojec've cer O c f focl length Lecture 8 -! 4
5 f rojec've cer O c y y c C[u o, v o ] x c x f focl length u o, v o offset Lecture 8 -! 5
6 f rojec've 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 Lecture 8 -! 6
7 f rojec've 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 Lecture 8 -! 7
8 f rojec've cer c O c ʹ α α cotθ β sn θ 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 Lecture 8 -! 8
9 f rojec've 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 roton, trnslon Lecture 8 -! 9
10 f rojec've cer R,T j w k w O w O c w ʹ M w K[ R T] w Internl (ntrnsc) preters Externl (extrnsc) preters f focl length u o, v o offset α β, non- squre pxels θ skew ngle R,T roton, trnslon Lecture 8 -!
11 rojec've cer ʹ K[ R T] w M w Internl (ntrnsc) preters Externl (extrnsc) preters Lecture 8 -!
12 Lecture 8 -! rojec've cer M w ʹ [ ] w T K R v u cot K o o sn θ β θ α α T T T R r r r z y x t t t T 4
13 Lecture 8 -! Gol of clbr'on M w ʹ [ ] w T K R v u cot K o o sn θ β θ α α T T T R r r r z y x t t t T 4 Este ntrnsc nd extrnsc preters fro or ulple ges
14 Wht we wll lern tody? Revew cer preters Affne cer odel (roble Set (Q4)) Cer clbron Vnshng ponts nd lnes Redng: [F] Chpter [HZ] Chpter 7, 8.6 Lecture 8 -! 4
15 Wek perspec've projec'on x' x y' y where f z ' gnfcon Relve scene depth s sll copred to ts dstnce fro the cer Lecture 8 -! 5
16 Orthogrphc (ffne) projec'on x' y' x y Dstnce fro center of projecon to ge plne s nfnte Lecture 8 -! 6
17 Lecture 8 -! 7 Affne cers [ ] T R K ' s K y x α α T R K M Affne cse rllel projecon trx T R K M y x s K o y o x α α rojecve cse Copred to
18 Lecture 8 -! 8 Reeber. rojecves: y x H y x b v t A y' x' p Affnes: y x H y x t A y' x'
19 Lecture 8 -! 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 forulon thn: Affne cers
20 Affne cers To recp: M cer trx ' u v A + b M ; M Ths noton s useful when we ll dscuss ffne structure fro oon [non- hoogeneous ge coordntes] [ A b] Lecture 8 -!
21 Affne cers Wek perspecve uch spler th. Accurte when object s sll nd dstnt. Most useful for recognon. nhole perspecve uch ore ccurte for scenes. Used n structure fro oon. Lecture 8 -!
22 Wek perspec've projec'on - exples The Kngx Eperor's Southern Inspec7on Tour (69-698) By Wng Hu You tube vdeo clck here Lecture 8 -!
23 Wek perspec've projec'on - exples Qngng Fes7vl by the Rversde Zhng Zedun ~9 AD Lecture 8 -!
24 Wht we wll lern tody? Revew cer preters Affne cer odel Cer clbron Vnshng ponts nd lnes Redng: [F] Chpter [HZ] Chpter 7, 8.6 Lecture 8 -! 4
25 Clbr'on roble Clbron rg j C n wth known posons n [O w, w, j w, k w ] p, p n known posons n the ge Gol: copute ntrnsc nd extrnsc preters Lecture 8 -! 5
26 Reeber the dgtl Mchelngelo project? Lecture 8 -! 6
27 Clbr'on roble Clbron rg j C How ny correspondences do we need? M hs unknown We need equons 6 correspondences would do t Lecture 8 -! 7
28 Clbr'on roble Clbron rg ge j C In prcce: user y need to look t the ge nd select the n>6 correspondences Lecture 8 -! 8
29 Lecture 8 -! 9 Clbr'on roble j C M p v u p M n pxels
30 Lecture 8 -! Clbr'on roble u ) ( v ) ( u v v u ) ( v ) ( u
31 Lecture 8 -! Clbr'on roble ) ( v ) ( u ) ( v ) ( u ) ( n n n v ) ( n n n u
32 Block Mtrx Mulplcon A A A A A B B B B B Wht s AB? AB A A B B + + A A B B A A B B + + A A B B Lecture 8 -!
33 Clbr'on 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 Lecture 8 -!
34 Hoogeneous M x N Lner Systes Mnuber of equons Nnuber of unknown A x Rectngulr syste (M>N) s lwys soluon To fnd non- zero soluon Mnze Ax under the constrnt x Lecture 8 -! 4
35 Clbr'on roble How do we solve ths hoogenous lner syste? Sngulr Vlue Decoposon (SVD) Lecture 8 -! 5
36 Clbr'on roble Copute SVD decoposon of U D V T n Lst colun of V gves Why? See pge 59 of Hrtley & Zssern M M p Lecture 8 -! 6
37 Lecture 8 -! 7 Extrc'ng cer preters A T T T A [ ] T K R ± ρ b b b b Ested vlues ) ( u o ρ ) ( v o ρ ( ) ( ) cos θ Intrnsc b v u cot K o o sn θ β θ α α ρ
38 Theore (Fugers, 99) [ T] [ K R KT ] [ A b] M K R A K α s cx c β y α f β f k; l Lecture 8 -! 8
39 Lecture 8 -! 9 Extrc'ng cer preters A T T T A [ ] T K R b b b b Ested vlues Intrnsc θ ρ α sn θ ρ β sn b f ρ
40 Lecture 8 -! 4 Extrc'ng cer preters Extrnsc ( ) r r ± r r r b K T ρ A T T T A [ ] T K R b b b b Ested vlues b ρ
41 Clbr'on Deo Cer Clbr7on Toolbox for Mtlb J. Bouguet [998- ] hxp:// Lecture 8 -! 4
42 Clbr'on Deo Lecture 8 -! 4
43 Clbr'on Deo Lecture 8 -! 4
44 Clbr'on Deo Lecture 8 -! 44
45 Clbr'on Deo Lecture 8 -! 45
46 Clbr'on Deo Lecture 8 -! 46
47 Clbr'on Deo Lecture 8 -! 47
48 Clbr'on Deo Lecture 8 -! 48
49 Wht we wll lern tody? Revew cer preters Affne cer odel Cer clbron Vnshng ponts nd lnes (roble Set (Q)) Redng: [F] Chpter [HZ] Chpter 7, 8.6 Lecture 8 -! 49
50 roper'es of rojec'on onts project to ponts Lnes project to lnes Lecture 8 -! 5
51 roper'es of rojec'on Angles re not preserved rllel lnes eet Vnshng pont Lecture 8 -! 5
52 Lecture 8 -! 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 Intersecng lnes x roof l l l lʹ l lʹ lʹ l lʹ ( l lʹ ) l ( l lʹ ) lʹ x y x lʹ x l x lʹ x s the ntersecng pont x Lecture 8 -! 5
54 Lecture 8 -! 54 onts t nfnty (del ponts) x, x x x x c b l ʹ ʹ c b l ʹ ʹ b c ) (c l l Let s ntersect two prllel lnes: Agree wth the generl de of two lnes ntersecng t nfnty l lʹ x x x
55 Lecture 8 -! 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 Lecture 8 -! 56 rojec've projec'ons of lnes t nfnty (D) l H l T ʹ 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 rojec've projec'ons of lnes t nfnty (D) horzon l hor T H l Are these two lnes prllel or not? Recognon helps reconstrucon! Huns hve lernt ths - Recognze the horzon lne - Mesure f the lnes eet t the horzon - f yes, these lnes re // Lecture 8 -! 57
58 Vnshng ponts ( del ponts n D) Vnshng ponts ponts where prllel lnes ntersect n D Ige of vnshng pont d ddrecon of the lne M K[ R T] v K d v C Lecture 8 -! 58
59 Horzon Sets of prllel lnes on the se plne led to collner vnshng ponts [The lne s clled the horzon for tht plne] horzon Lecture 8 -! 59
60 Horzon n l horz C T n K l horz Lecture 8 -! 6
61 Applc'on These trnsforons re used n sngle vew etrology Crns & Zssern, 99 Lecture 8 -! 6
62 Applc'on these trnsforons re used n sngle vew etrology Crns & Zssern, 99 Lecture 8 -! 6
63 Applc'on these trnsforons re used n sngle vew etrology Crns & Zssern, 99 L Trnt' (46) Frenze, Snt Mr Novell; by Mscco (4-48) Lecture 8 -!6
64 Lecture 8 -! 64
65 Applc'on these trnsforons re used n sngle vew etrology Hoe et l, 5 hxp:// wre.htl Lecture 8 -! 65
66 Applc'on these trnsforons re used n sngle vew etrology Sxen, Sun, Ng, 5 A softwre: MkeD Convert your ge nto d odel hxp://ked.stnford.edu/ hxp://ked.stnford.edu/ges/vewd/85 hxp://ked.stnford.edu/ges/vewd/9?noforwrdtrue hxp://ked.stnford.edu/ges/vewd/8 Lecture 8 -! 66
67 Wht we hve lerned tody Revew cer preters Affne cer odel (roble Set (Q4)) Cer clbron Vnshng ponts nd lnes (roble Set (Q)) Redng: [F] Chpter [HZ] Chpter 7, 8.6 Lecture 8 -! 67
68 Suppleentry Mterls Lecture 8 -! 68
69 Degenercy nd dstoron n rel- world cer clbron Lecture 8 -! 69
70 Degenerte cses s cnnot le on the se plne! onts cnnot le on the ntersecon curve of two qudrc surfces Lecture 8 -! 7
71 Rdl Dstor'on Cused by perfect lenses Devons re ost noceble for rys tht pss through the edge of the lens No dstorton n cushon Brrel Lecture 8 -! 7
72 Lecture 8 -! 7 Rdl Dstor'on p v u M λ λ d v v c u b v u d + + u ± p p κ p d λ olynol funcon Dstoron coeffcent To odel rdl behvor
73 Lecture 8 -! 7 Rdl Dstor'on 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 equons
74 Generl Clbr'on roble X f () f( ) s nonlner esureent preter - Newton Method - Levenberg- Mrqurdt Algorth Iterve, strts fro nl soluon My be slow f nl soluon fr fro rel soluon Ested soluon y be funcon of the nl soluon Newton requres the coputon of J, H Levenberg- Mrqurdt doesn t requre the coputon of H Lecture 8 -! 74
75 Generl Clbr'on roble X f () f( ) s nonlner esureent preter A possble lgorth. Solve lner prt of the syste to fnd pproxted soluon. Use ths soluon s nl condon for the full syste. Solve full syste (ncludng dstoron) usng Newton or L.M. Lecture 8 -! 75
76 Generl Clbr'on roble X f () f( ) s nonlner esureent preter Typcl ssupons for copung nl condon : - zero- skew, squre pxel - u o, v o known center of the ge - no dstoron Just este f nd R, T Lecture 8 -! 76
77 Lecture 8 -! 77 Ts s clbr'on technque. Este nd frst: v u p λ How to do tht? d v u Hnt: slope v u
78 Lecture 8 -! 78 Ts s clbr'on technque. Este nd frst: v u p λ ) ( ) ( u v ) ( ) ( u v ) ( ) ( n n n n u v Q n n v u ) ( ) ( ) ( ) (
79 Lecture 8 -! 79 Ts s clbr'on technque. Once tht nd re ested, este : v u p λ s non lner funcon of λ There re soe degenerte confgurons for whch nd cnnot be coputed
Lecture 8: Camera Calibration
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