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
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
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 '
Coo asfoaios Oigial Tasfoed Taslaio Roaio Scalig Affie Pespecive Slide cedi (e few slides): A. Efos ad/o S. Seiz
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
Scalig No-uifo scalig: diffee scalas pe copoe: 2,.5
Scalig Scalig opeaio: ' ' a b O, i ai fo: ' ' a b scalig ai S
2-D Roaio (, ) (, ) = cos() - si() = si() + cos()
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
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
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
2D iage asfoaios (efeece able) Hoogaph Szeliski 2.
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)
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 2 22 23 3 32 33 z z
Hoogeeous coodiaes Allows ai fo fo 3D asfoaios Take oaio = Now add aslaio = 2 3 2 22 23 3 32 33 2 3 2 22 23 3 32 33 + 2 3
Hoogeeous coodiaes Allows ai fo fo 3D asfoaios Take oaio 2 3 = 2 22 23 3 32 33 Now add aslaio wih hoogeeous coodiae = 2 3 2 22 23 2 3 32 33 3
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: (,,,)
Iisic ad eisic aices K R 33 32 3 23 22 2 3 2 z v f u s f v u w z Has
Hoogeeous coodiaes Pojecive Poi becoes a lie To hoogeeous Fo hoogeeous Sog Ho Ah
How o calibae he caea? (also called caea esecioig ) * * * * * * * * * * * * w wv wu K R Jaes Has = M Liea leas-squaes egessio!
Siple eaple: Fiig a lie
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)
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
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
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
Ae hese asfoaios eough?
Wold vs Caea coodiaes Jaes Has
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 2 3 2 22 32 M 3 23 33 4 24 34 Ukow Caea Paaees
How do we calibae a caea? Kow 2d iage coods Kow 3d wold locaios 88 24 43 23 27 97 886 347 745 32 943 28 476 59 49 24 37 335 783 52 235 427 665 429 655 362 427 333 42 45 746 35 434 45 525 234 76 38 62 87 32.747 39.4 3.86 35.796 3.649 3.356 37.694 32.358 3.48 3.49 37.86 29.298 3.937 3.5 29.26 3.22 37.572 3.682 37.6 36.876 28.66 39.37 32.49 3.23 37.435 3.5 29.38 38.253 36.3 28.88 36.65 39.3 28.95 38.69 36.83 29.89 39.67 38.834 29.29 38.255 39.955 29.267 37.546 38.63 28.963 3.36 39.26 28.93 37.58 38.75 29.69 39.95 3.262 29.99 32.6 3.772 29.8 3.988 32.79 3.54 Jaes Has
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
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
34 33 32 3 24 23 22 2 4 3 2 s sv su 4 3 2 su 24 23 22 2 sv 34 33 32 3 s Kow 3d locaios Kow 2d iage coods Ukow Caea Paaees 34 33 32 3 4 3 2 u 34 33 32 3 24 23 22 2 v Jaes Has Two equaios pe 3D poi coespodece Fis, wok ou whee,, pojecs o ude cadidae M.
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 2 22 32 3 2 3 3 23 33 2 32 22 32 3 32 33 34) 3 32 33 34) 4 24 34 3 33 23 33 Kow 3d locaios 4 ( u ( v 3u 32u 33u 34u 2 3 4 3v 32v 33v 34v 2 22 23 24 2 34 24 34 2 22 3 23 4 24
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 2 22 32 3 23 33 4 24 34 Kow 3d locaios 3u 32u 33u 34u 2 3 4 3v 32v 33v 34v 2 22 23 24 2 3 4 3u 32u 33u 34u v v v v 2 22 23 24 3 32 33 34
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 33 32 3 24 23 22 2 4 3 2 A=b fo M = A\; M = [M;]; M = eshape(m,[],3)'; 34 33 32 3 24 23 22 2 4 3 2 s sv su Kow 3d locaios Kow 2d iage coods Jaes Has Ukow Caea Paaees
34 33 32 3 24 23 22 2 4 3 2 s sv su Kow 3d locaios Kow 2d iage coods O, solve fo s eies usig oal liea leas-squaes. Mehod 2 34 33 32 3 24 23 22 2 4 3 2 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
How do we calibae a caea? Kow 2d iage coods Kow 3d wold locaios 88 24 43 23 27 97 886 347 745 32 943 28 476 59 49 24 37 335 783 52 235 427 665 429 655 362 427 333 42 45 746 35 434 45 525 234 76 38 62 87 32.747 39.4 3.86 35.796 3.649 3.356 37.694 32.358 3.48 3.49 37.86 29.298 3.937 3.5 29.26 3.22 37.572 3.682 37.6 36.876 28.66 39.37 32.49 3.23 37.435 3.5 29.38 38.253 36.3 28.88 36.65 39.3 28.95 38.69 36.83 29.89 39.67 38.834 29.29 38.255 39.955 29.267 37.546 38.63 28.963 3.36 39.26 28.93 37.58 38.75 29.69 39.95 3.262 29.99 32.6 3.772 29.8 3.988 32.79 3.54 Jaes Has
Kow 2d iage coods Kow 3d wold locaios s poi 32.747 39.4 88 24 43 23 27 97 886 347 745 32 943 28 476 59 49 24 37 335 3.86 (u, v ) (,, ) 32.747 39.4 3.86 8832.747 2432.747 u v 32.747 39.4 3.86 35.796 3.649 3.356 37.694 32.358 3.48 3.49 37.86 29.298 3.937 3.5 29.26 3.22 37.572 3.682 37.6 36.876 28.66 39.37 32.49 3.23 37.435 3.5 29.38.. 8839.4 883.86 88 2439.4 243.86 24 u u u v v v Pojecio eo defied b wo equaios oe fo u ad oe fo v 2 3 4 2 22 23 24 3 32 33 34
Kow 2d iage coods Kow 3d wold locaios 88 24 43 23 27 97 886 347 745 32 943 28 476 59 49 24 37 335 32.747 39.4 3.86 35.796 3.649 3.356 37.694 32.358 3.48 3.49 37.86 29.298 3.937 3.5 29.26 3.22 37.572 3.682 37.6 36.876 28.66 39.37 32.49 3.23 37.435 3.5 29.38.. 2 d poi (u 2, v 2 ) ( 2, 2, 2 ) 32.747 35.796 Pojecio eo defied b wo equaios oe fo u ad oe fo v 39.4 3.86 8832.747 8839.4 883.86 32.747 39.4 3.86 2432.747 2439.4 243.86 3.649 3.356 4335.796 433.649 433.356 35.796 3.649 3.356 2335.796 233.649 433.356 u v u v u v 88 24 43 23 u v 2 3 4 2 22 23 24 3 32 33 34
How a pois do I eed o fi he odel? K R 33 32 3 23 22 2 3 2 z v u s v u w z 5 6 Degees of feedo? Thik 3: - Roaio aoud - Roaio aoud - Roaio aoud z
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 2 22 23 3 32 33 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.
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
Ca we facoize M back o K [R T]? es! We ca diecl solve fo he idividual eies of K [R T]. Jaes Has
a = h colu of A Jaes Has
Jaes Has
Jaes Has
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
Recoveig he caea cee * * * * * * * * * * * * s sv su K R 33 32 3 23 22 2 3 2 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
Esiae of caea cee.486 -.3645 -.685 -.44 -.9437 -.42.682.699.677 -.77.2543 -.6454 -.279.8635 -.457 -.3645 -.792.37.738.6382 -.58.332.3464.3377.337.89 -.43.242 -.4799.292.69.83 -.48.292 -.9 -.2992.529 -.575.46 -.4527.576 -.49.2598 -.5282.9695.382 -.682.2856.478.424 -.2 -.95.295.282 -.28.889 -.848.5255 -.9442 -.583 -.3759.45.3445.324 -.7975.37 -.826 -.4329 -.45 -.2774 -.475 -.772 -.2667 -.549 -.784 -.4.993 -.2854 -.24 -.432.243 -.53 -.748 -.384 -.248.878 -.96 -.263 -.765 -.5792 -.936.3237.797.27.389.5786 -.887.2323.442.456
Oieed ad Taslaed Caea R j w k w O w i w
ONE DIFFICULT EAMPLE
Eik Johasso The Achiec