Cameras and World Geometry

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2 Caeas ad Wold Geoe How all is his woa? How high is he caea? Wha is he caea oaio w. wold? Which ball is close? Jaes Has

3 Thigs o eebe Has Pihole caea odel ad caea (pojecio) ai Hoogeeous coodiaes allow pojecio as a ai uliplicaio K R Model appoiaes a caea Les disoio Apeue

4 Paaeic (global) asfoaios T p = (,) p = (, ) Tasfoaio T is a coodiae-chagig achie: p = T(p) Wha does i ea ha T is global? T is he sae fo a poi p T ca be descibed b jus a few ubes (paaees) Fo liea asfoaios, we ca epese T as a ai p = Tp ' T '

5 Coo asfoaios Oigial Tasfoed Taslaio Roaio Scalig Affie Pespecive Slide cedi (e few slides): A. Efos ad/o S. Seiz

6 Scalig Scalig a coodiae eas uliplig each of is copoes b a scala Uifo scalig eas his scala is he sae fo all copoes: 2

7 Scalig No-uifo scalig: diffee scalas pe copoe: 2,.5

8 Scalig Scalig opeaio: ' ' a b O, i ai fo: ' ' a b scalig ai S

9 2-D Roaio (, ) (, ) = cos() - si() = si() + cos()

10 2-D Roaio This is eas o capue i ai fo: ' ' cos si si cos Eve hough si() ad cos() ae oliea fucios of, is a liea cobiaio of ad is a liea cobiaio of ad Wha is he ivese asfoaio? Roaio b Fo oaio aices R T R R

11 Basic 2D asfoaios Taslae Roae Shea Scale ' ' cos si si cos ' ' s s ' ' f e d c b a Affie Affie is a cobiaio of aslaio, scale, oaio, ad shea

12 Affie Tasfoaios Affie asfoaios ae cobiaios of Liea asfoaios, ad Taslaios Popeies of affie asfoaios: Lies ap o lies Paallel lies eai paallel Raios ae peseved Closed ude coposiio f e d c b a ' ' f e d c b a o

13 2D iage asfoaios (efeece able) Hoogaph Szeliski 2.

14 Pojecive Tasfoaios Pojecive asfoaios ae cobos of Affie asfoaios, ad Pojecive waps ' ' w' a d g b e h c f i w Popeies of pojecive asfoaios: Lies ap o lies Paallel lies do o ecessail eai paallel Raios ae o peseved Closed ude coposiio Models chage of basis Pojecive ai is defied up o a scale (8 DOF)

15 Caea (pojecio) ai R, Slide Cedi: Savaese j w k w O w i w K R : Iage Coodiaes: (u,v,) K: Iisic Mai (33) R: Roaio (33) : Taslaio (3) : Wold Coodiaes: (,,,) u f s u 2 3 w v f v z z

16 Hoogeeous coodiaes Allows ai fo fo 3D asfoaios Take oaio = Now add aslaio =

17 Hoogeeous coodiaes Allows ai fo fo 3D asfoaios Take oaio 2 3 = Now add aslaio wih hoogeeous coodiae =

18 Caea (pojecio) ai Slide Cedi: Savaese R, j w k w O w i w K R Eisic Mai : Iage Coodiaes: (u,v,) K: Iisic Mai (33) R: Roaio (33) : Taslaio (3) : Wold Coodiaes: (,,,)

19 Iisic ad eisic aices K R z v f u s f v u w z Has

20 Hoogeeous coodiaes Pojecive Poi becoes a lie To hoogeeous Fo hoogeeous Sog Ho Ah

21 How o calibae he caea? (also called caea esecioig ) * * * * * * * * * * * * w wv wu K R Jaes Has = M Liea leas-squaes egessio!

22 Siple eaple: Fiig a lie

23 Leas squaes lie fiig Daa: (, ),, (, ) Lie equaio: i = i + b Fid (, b) o iiize 2 2 A Ap A T T dp de 2 2 i i i b b E i i i b E 2 ) ( ( i, i ) =+b A A A p A Ap A T T T T Malab: p = A \ ; Modified fo S. Lazebik ) ( ) ( ) ( 2 Ap Ap Ap T T T 2 Ap (Closed fo soluio)

24 Eaple: solvig fo aslaio A A 2 A 3 B B 2 B 3 Give ached pois i {A} ad {B}, esiae he aslaio of he objec A i A i B i B i

25 Eaple: solvig fo aslaio A A 2 A 3 B B 2 B 3 Leas squaes seup A i A i B i B i (, ) A B A B A B A B

26 Eaple: discoveig o/as/scale A A 2 A 3 Give ached pois i {A} ad {B}, esiae he asfoaio ai A i A i B i B i d c b a

27 Ae hese asfoaios eough?

28 Wold vs Caea coodiaes Jaes Has

29 Calibaig he Caea Jaes Has Use a scee wih kow geoe Coespod iage pois o 3d pois Ge leas squaes soluio (o o-liea soluio) Kow 2d iage coods Kow 3d wold locaios su sv s M Ukow Caea Paaees

30 How do we calibae a caea? Kow 2d iage coods Kow 3d wold locaios Jaes Has

31 Wha is leas squaes doig? Give 3D poi evidece, fid bes M which iiizes eo bewee esiae (p ) ad kow coespodig 2D pois (p). Eo bewee M esiae ad kow pojecio poi. f.. P p ude M.. u v Caea cee u p v p = disace fo iage cee

32 Wha is leas squaes doig? Bes M occus whe p = p, o whe p p = Fo hese equaios fo all poi evidece Solve fo odel via closed-fo egessio Eo bewee M esiae ad kow pojecio poi p ude M.. u. f v u p v. Caea cee p = disace fo iage cee. P

33 s sv su su sv s Kow 3d locaios Kow 2d iage coods Ukow Caea Paaees u v Jaes Has Two equaios pe 3D poi coespodece Fis, wok ou whee,, pojecs o ude cadidae M.

34 Ukow Caea Paaees Kow 2d iage coods Ne, eaage io fo whee all M coefficies ae idividuall saed i es of,,,u,v. -> Allows us o fo lsq ai. su sv s 2 3 u v ) ) Kow 3d locaios 4 ( u ( v 3u 32u 33u 34u v 32v 33v 34v

35 Ukow Caea Paaees Kow 2d iage coods Ne, eaage io fo whee all M coefficies ae idividuall saed i es of,,,u,v. -> Allows us o fo lsq ai. su sv s Kow 3d locaios 3u 32u 33u 34u v 32v 33v 34v u 32u 33u 34u v v v v

36 Fiall, solve fo s eies usig liea leas squaes Mehod v u v u v v v u u u v v v u u u A=b fo M = A\; M = [M;]; M = eshape(m,[],3)'; s sv su Kow 3d locaios Kow 2d iage coods Jaes Has Ukow Caea Paaees

37 s sv su Kow 3d locaios Kow 2d iage coods O, solve fo s eies usig oal liea leas-squaes. Mehod v v v v u u u u v v v v u u u u [U, S, V] = svd(a); M = V(:,ed); M = eshape(m,[],3)'; A= fo Jaes Has Ukow Caea Paaees

38 How do we calibae a caea? Kow 2d iage coods Kow 3d wold locaios Jaes Has

39 Kow 2d iage coods Kow 3d wold locaios s poi (u, v ) (,, ) u v u u u v v v Pojecio eo defied b wo equaios oe fo u ad oe fo v

40 Kow 2d iage coods Kow 3d wold locaios d poi (u 2, v 2 ) ( 2, 2, 2 ) Pojecio eo defied b wo equaios oe fo u ad oe fo v u v u v u v u v

41 How a pois do I eed o fi he odel? K R z v u s v u w z 5 6 Degees of feedo? Thik 3: - Roaio aoud - Roaio aoud - Roaio aoud z

42 How a pois do I eed o fi he odel? K R Degees of feedo? 5 6 u s u 2 3 w v v z z M is 34, so 2 ukows, bu pojecive scale abigui deg. feedo. Oe equaio pe ukow -> 5 /2 poi coespodeces deeies a soluio (e.g., eihe u o v). Moe ha 5 /2 poi coespodeces -> ovedeeied, a soluios o M. Leas squaes is fidig he soluio ha bes saisfies he ovedeeied sse. Wh use oe ha 6? Robusess o eo i feaue pois.

43 Calibaio wih liea ehod Advaages Eas o foulae ad solve Povides iiializaio fo o-liea ehods Disadvaages Does diecl give ou caea paaees Does odel adial disoio Ca ipose cosais, such as kow focal legh No-liea ehods ae pefeed Defie eo as diffeece bewee pojeced pois ad easued pois Miiize eo usig Newo s ehod o ohe o-liea opiizaio Jaes Has

44 Ca we facoize M back o K [R T]? es! We ca diecl solve fo he idividual eies of K [R T]. Jaes Has

45 a = h colu of A Jaes Has

46 Jaes Has

47 Jaes Has

48 Ca we facoize M back o K [R T]? es! We ca also use RQ facoizaio (o QR) R i RQ is o oaio ai R; cossed aes! R (igh diagoal) is K Q (ohogoal basis) is R. T, he las colu of [R T], is iv(k) * las colu of M. Bu ou eed o do a bi of pos-pocessig o ake sue ha he aices ae valid. See hp://ksiek.gihub.io/22/8/4/decopose/ Jaes Has

49 Recoveig he caea cee * * * * * * * * * * * * s sv su K R z v u s v u w z This is o he caea cee C. I is RC, as he poi is oaed befoe,, ad z ae added This is K Q So K - 4 is So we eed -R - K - 4 o ge C. Q is K R. So we jus eed -Q - 4 Jaes Has 4

50 Esiae of caea cee

51 Oieed ad Taslaed Caea R j w k w O w i w

52 ONE DIFFICULT EAMPLE

53 Eik Johasso The Achiec

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