Camera Models class 8

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Dortha Thompson
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1 Camea Models class 8 Mulile View Geomey Com Mac ollefeys

2 Mulile View Geomey couse schedule (subjec o change) Jan. 7, 9 Ino & moivaion ojecive 2D Geomey Jan. 4, 6 (no class) ojecive 2D Geomey Jan. 2, 23 ojecive 3D Geomey (no class) Jan. 28, 3 aamee Esimaion aamee Esimaion Feb. 4, 6 Algoihm Evaluaion Camea Models Feb., 3 Camea Calibaion Single View Geomey Feb. 8, 2 Eiola Geomey 3D econsucion Feb. 25, 27 Fund. Mai Com. Sucue Com. Ma. 4, 6 lanes & Homogahies Tifocal Tenso Ma. 8, 2 Thee View Reconsucion Mulile View Geomey Ma. 25, 27 MulileView Reconsucion Bundle adjusmen A., 3 Auo-Calibaion aes A. 8, Dynamic SfM aes A. 5, 7 Cheialiy aes A. 22, 24 Dualiy ojec Demos

RMS esimaion eo fo ML esimao e es / 2 2 E ˆ / N σ E ˆ 2 / N / 2 σ ( d / N )

3 N measuemens (indeenden Gaussian noise σ 2 ) model wih d essenial aamees (use sd and s(n-d)) (i) RMS esidual eo fo ML esimao e es ( d / N ) / 2 (ii) RMS esimaion eo fo ML esimao e es / 2 2 E ˆ / N σ E ˆ 2 / N / 2 σ ( d / N ) / 2 n S M Eo in wo images e es n + 4 σ 2n / 2 d 8 + 2n and N 4n e es σ n 4 2n / 2

4 Fowad oagaion of covaiance Σ Backwad oagaion of covaiance Σ ( ) J T Σ J - Ove-aameeizaion Σ J T Σ J ( T J Σ J) + Mone-Calo esimaion of covaiance A f J v f - η

5 Eamle: σ iel Σ.5cm (Cimisi 97)

6 Single view geomey Camea model Camea calibaion Single view geom.

8 inhole camea model T T Z fy Z f Z Y ) /, / ( ),, ( a Z Y f f Z fy f Z Y a

9 inhole camea model Z Y f f Z fy f Z Y f f Z fy f [ ] I,), diag( f f

10 incial oin offse T y T Z fy Z f Z Y ) /, / ( ),, ( + + a incial oin T y ), ( + + Z Y f f Z Z fy Z f Z Y y y a

11 incial oin offse + + Z Y f f Z Z fy Z f y [ ] cam I y f f calibaion mai

12 Camea oaion and anslaion ~ ( - C ~ ) cam R ~ RC ~ R Y R RC ~ cam Z [ ] I [ cam R I C ~ ] R RC ~ [ ]

13 CCD camea y y α α y y f f m m

14 Finie ojecive camea α s α y y R I [ ] C ~ dof (5+3+3) non-singula decomose in,r,c? [ M ] [ ] ( ) 4, R RQ M M 4 {finie cameas}{ 34 de M } If ank 3, bu ank M<3, hen cam a infiniy C ~

15 Camea anaomy Camea cene Column oins incial lane Ais lane incial oin incial ay

16 Camea cene null-sace camea ojecion mai C C is camea cene Image of camea cene is (,,) T, i.e. undefined M Finie cameas: C 4 d Infinie cameas: C,Md

17 Column vecos [ ] [ ] Image oins coesonding o,y,z diecions and oigin

18 Row vecos 3 2 Z Y y T T T 3 2 Z Y w y T T T noe:, 2 deenden on image eaameizaion

19 The incial oin incial oin (,,,) 3 ˆ ˆ 3 3 Mm

20 The incial ais veco veco defining fon side of camea 3 m [ ] camcam I ( ) ( ) T cam v de M m 3,, a k cam cam [ ] [ M ] kr I C ~ a k cam cam 4 v a k 4 v (diecion unaffeced) v a de ( ) 3 4 k M km k v

21 Acion of ojecive camea on oin Fowad ojecion [ M ] D Md D 4 Back-ojecion C + + T ( T ) ( λ) + λc + ( ) λ (seudo-invese) - d M - - M - M μ + D C 4 M - + I ( μ - ) 4

22 Deh of oins w 3 T 3 T ( C) m ( ) 3 T (C) ~ C ~ (do oduc) de M > ; m 3 If, hen m 3 uni veco in osiive diecion deh sign(dem) w ( ;) 3 T m (,Y,Z,T) T

23 Camea mai decomosiion Finding he camea cene C (use SVD o find null-sace) de( [ 2,3, 4] ) Y de( [,3, 4] ) Z de( [,, ]) T de( [,, ]) 2 4 Finding he camea oienaion and inenal aamees M R (use RQ ecomosiion) 2 3

24 When is skew non-zeo? α s α y γ acan(/s) fo CCD/CMOS, always s Image fom image, s ossible (non coinciding incial ais) esuling camea: H

25 Euclidean vs. ojecive geneal ojecive ineeaion [ 3 3 homogahy] [ 4 4 homogahy] Meaningfull decomosiion in,r, equies Euclidean image and sace Camea cene is sill valid in ojecive sace incial lane equies affine image and sace incial ay equies affine image and Euclidean sace

26 Cameas a infiniy Camea cene a infiniy de M Affine and non-affine cameas Definiion: affine camea has 3T (,,,)

27 Affine cameas

28 Affine cameas R I C ~ d 3T [ ] C ~ T 2T 3T T 2T 3T C ~ C ~ C ~ T 2T 3T T 2T 3T modifying 34 coesonds o moving along incial ay ( ) 3 T T C ~ ( ) 3 ( ) 2T 2T C ~ - C ~ 3 3T C ~ - - d C ~

29 Affine cameas 3T 2T 2T T T 3T 2T 2T T T / C ~ C ~ C ~ C ~ / / d d d d d d d d d d now adjus zoom o comensae 2T 2T T T C ~ C ~ lim d

30 Eo in emloying affine cameas + β α 2 Δ + + β α 3 2 oin on lane aallel wih incial lane and hough oigin, hen geneal oins + Δ ~ ~ oj d y affine ~ ~ d y oj affine

31 Affine imaging condiions Δ ( ) affine - oj oj - d Aoimaion should only cause small eo. Δ much smalle han d 2. oins close o incial oin (i.e. small field of view)

32 Decomosiion of 22 ~ R ~ ~ d ~ R ~ ~ 22 - d absob d in R ~ ~ ~ ~ ~ R ~ ~ ~ R ~ R ~ ~ ~ R ~ alenaives, because 8dof (3+3+2), no moe

33 Summay aallel ojecion canonical eesenaion 2 2 calibaion mai incial oin is no defined

34 A hieachy of affine cameas Ohogahic ojecion Scaled ohogahic ojecion R H 2 T T k / 2 T T (5dof) (6dof)

35 A hieachy of affine cameas Weak esecive ojecion α / k T T α y 2 (7dof)

36 A hieachy of affine cameas Affine camea α (8dof) / k T s T A α y 2 A m m [ 3 3 affine] [ 4 4 affine]. Affine cameacamea wih incial lane coinciding wih Π 2. Affine camea mas aallel lines o aallel lines 3. No cene of ojecion, bu diecion of ojecion A D (oin on Π ) m m m m 2 3 A

37 Summay of oeies of a ojecive Camea

38 ushboom cameas (dof) (,Y,,T) T T T (, y, w) (, y / w) Saigh lines ae no maed o saigh lines! (ohewise i would be a ojecive camea)

39 Line cameas y Y Z (5dof) Null-sace C yields camea cene Also decomosiion R [ I ~ c]

40 Ne class: Camea calibaion

Today - Lecture 13. Today s lecture continue with rotations, torque, Note that chapters 11, 12, 13 all involve rotations

Today - Lecture 13. Today s lecture continue with rotations, torque, Note that chapters 11, 12, 13 all involve rotations Today - Lecue 13 Today s lecue coninue wih oaions, oque, Noe ha chapes 11, 1, 13 all inole oaions slide 1 eiew Roaions Chapes 11 & 1 Viewed fom aboe (+z) Roaional, o angula elociy, gies angenial elociy