Lecture Caera Models Caera Calbraton rofessor Slvo Savarese Coputatonal Vson and Geoetry Lab Slvo Savarese Lecture - - Jan- 5
Lecture Caera Models Caera Calbraton Recap of caera odels Caera calbraton proble Caera calbraton wth radal dstorton Exaple Readng: [F] Chapter Geoetrc Caera Calbraton [HZ] Chapter 7 Coputaton of Caera Matrx Soe sldes n ths lecture are courtesy to rofs. J. once, F-F L Slvo Savarese Lecture - 4- Jan- 5
rojectve caera nhole perspectve projecton f f focal length
rojectve caera nhole perspectve projecton f y y c x c f focal length u o, v o offset (note a dfferent conventon w.r.t. lecture ) C [u o, v o ] x
rojectve caera f Unts: k,l [pxel/] α, β f [] Non-square pxels [pxel] f focal length u o, v o offset α, β non-square pxels
rojectve caera f y y c θ C [u o, v o ] x c x f focal length u o, v o offset α, β non-square pxels θ skew angle
rojectve caera f ( ( ( ( '( α α cotθ u o β snθ v o K has 5 degrees of freedo ) + ( + ( + ( + ( * + ' x y z ) + + + + * f focal length u o, v o offset α, β non-square pxels θ skew angle
rojectve caera j w R, f O w k w ' R 4 4 w f focal length u o, v o offset α, β non-square pxels θ skew angle R, rotaton, translaton
rojectve caera j w R, f O w k w M w K[ R ] w Internal paraeters External paraeters f focal length u o, v o offset α, β non-square pxels θ skew angle R, rotaton, translaton
rojectve caera j w R, f O w k w ' M K R w W W W W w4 4 M E ' E ( w, w ) w w
heore (Faugeras, 99) [ R ] [ K R K ] [ A b] M K [Eq.; lect. ] A a a a
Exercse R, j w f k w O w w à M K R ' E ( w w, w w ) K I ( f x w z w, f y w z w ) f f w x w y w z w
rojectve caera p q r f O Q R
Weak perspectve projecton When the relatve scene depth s sall copared to ts dstance fro the caera p f R R_ q r Q Q_ O zo _ π
Weak perspectve projecton f zo p R_ q r Q Q_ O R x' f ' z x y' f ' z y x' f ' z x y' f ' z y _ π Magnfcaton
Weak perspectve projecton f zo p R_ q r Q Q_ O π R _ rojectve (perspectve) Weak perspectve M K[ R ] A b v à M A b
W W W ), ( w w E M w b A M agnfcaton M w v b A M W W W W ), ( w w w w E erspectve Weak perspectve
Orthographc (affne) projecton Dstance fro center of projecton to age plane s nfnte x' f ' z x y' f ' z y x' x y' y
ros and Cons of hese Models Weak perspectve uch spler ath. Accurate when object s sall and dstant. Most useful for recognton. nhole perspectve uch ore accurate for scenes. Used n structure fro oton or SLAM.
Weak perspectve projecton he Kangx Eperor's Southern Inspecton our (69-698) By Wang Hu
Weak perspectve projecton he Kangx Eperor's Southern Inspecton our (69-698) By Wang Hu
Lecture Caera Calbraton Recap of caera odels Caera calbraton proble Caera calbraton wth radal dstorton Exaple Readng: [F] Chapter Geoetrc Caera Calbraton [HZ] Chapter 7 Coputaton of Caera Matrx Soe sldes n ths lecture are courtesy to rofs. J. once, F-F L Slvo Savarese Lecture - 4- Jan- 5
rojectve caera v u cot K o o snθ β θ α α R r r r z y x t t t 4 M w [ ] w K R Internal paraeters External paraeters
Goal of calbraton M w [ ] w K R Estate ntrnsc and extrnsc paraeters fro or ultple ages Change notaton: w p v u cot K o o snθ β θ α α R r r r z y x t t t 4
Calbraton roble Calbraton rg j C n wth known postons n [O w, w,j w,k w ]
Calbraton roble Calbraton rg age p j C n wth known postons n [O w, w,j w,k w ] p, p n known postons n the age Goal: copute ntrnsc and extrnsc paraeters
Calbraton roble Calbraton rg age p j C How any correspondences do we need? M has unknown We need equatons 6 correspondences would do t
Calbraton roble Calbraton rg age p j C In practce, usng ore than 6 correspondences enables ore robust results
Calbraton roble j C M v u p M n pxels [Eq. ]
Calbraton roble u ) ( v ) ( u v v u ) ( v u ( ) [Eq. ] [Eqs. ]
Calbraton roble ) ( v ) ( u ) ( v ) ( u ) ( n n n v ) ( n n n u [Eqs. ]
Block Matrx Multplcaton B B B B B A A A A A What s AB? + + + + B A B A B A B A B A B A B A B A AB
Calbraton roble u ( ) + v ( ) + known unknown [Eq. 4] u v n n ( n ) + n ( n ) + n Hoogenous lnear syste u v n un n n un n ' def x4 n x def 4x x
Hoogeneous M x N Lnear Systes Mnuber of equatons n Nnuber of unknown M N Rectangular syste (M>N) s always a soluton o fnd non-zero soluton Mnze under the constrant
Calbraton roble How do we solve ths hoogenous lnear syste? Va SVD decoposton
Calbraton roble SVD decoposton of U D V n Last colun of V gves Why? See pag 59 of HZ def M
Extractng caera paraeters M ρ
Extractng caera paraeters A A a a a [ ] K R a ± ρ b b b b Estated values ( ) a a ρ u o ) ( v o a a ρ ( ) ( ) cos a a a a a a a a θ Intrnsc b v u cot K o o snθ β θ α α ρ M Box
heore (Faugeras, 99)
Extractng caera paraeters A a a a [ ] K R b b b b Estated values Intrnsc θ ρ α sn a a θ ρ β sn a a ρ A b
Extractng caera paraeters A A a a a b [ ] K R b b b b Estated values Extrnsc ( ) a a a a r a a r ± r r r b K ρ ρ
Degenerate cases s cannot le on the sae plane onts cannot le on the ntersecton curve of two quadrc surfaces
Lecture Caera Calbraton Recap of projectve caeras Caera calbraton proble Caera calbraton wth radal dstorton Exaple Readng: [F] Chapter Geoetrc Caera Calbraton [HZ] Chapter 7 Coputaton of Caera Matrx Soe sldes n ths lecture are courtesy to rofs. J. once, F-F L Slvo Savarese Lecture - 4- Jan- 5
Radal Dstorton Iage agnfcaton (n)decreases wth dstance fro the optcal axs Caused by perfect lenses Devatons are ost notceable for rays that pass through the edge of the lens No dstorton n cushon Barrel
Radal Dstorton p v u M λ λ d v v u c v b u a d + + u o odel radal behavor ± p p κ p d λ olynoal functon Dstorton coeffcent [Eq. 5] [Eq. 6] S λ Iage agnfcaton decreases wth dstance fro the optcal center
v u p Q q q q q q q q p v u M λ λ Q v u q q q q Is ths a lnear syste of equatons? Radal Dstorton No why? [Eqs.7]
General Calbraton roble u v q q q q n X f (Q) [Eq.8] paraeters easureents f( ) s the nonlnear appng -Newton Method -Levenberg-Marquardt Algorth Iteratve, starts fro ntal soluton May be slow f ntal soluton far fro real soluton Estated soluton ay be functon of the ntal soluton Newton requres the coputaton of J, H Levenberg-Marquardt doesn t requre the coputaton of H
General Calbraton roble u v q q q q n X f (Q) [Eq.8] paraeters easureents f( ) s the nonlnear appng A possble algorth. Solve lnear part of the syste to fnd approxated soluton. Use ths soluton as ntal condton for the full syste. Solve full syste usng Newton or L.M.
ypcal assuptons: - zero-skew, square pxel - u o, v o known center of the age General Calbraton roble X f (Q) easureent paraeter f( ) s nonlnear q q q q v u
Can we estate and and gnore the radal dstorton? Radal Dstorton v u p λ d v u Hnt: slope v u q q q q
Estatng and Radal Dstorton v u p λ ) ( ) ( u v ) ( ) ( u v ) ( ) ( n n n n u v L n n sa [87] v u ) ( ) ( ) ( ) ( Get and by SVD [Eq.9] [Eq.] [Eq.]
Once that and are estated Radal Dstorton v u p λ s non lnear functon of λ here are soe degenerate confguratons for whch and cannot be coputed,,
Lecture Caera Calbraton Recap of projectve caeras Caera calbraton proble Caera calbraton wth radal dstorton Exaple Readng: [F] Chapter Geoetrc Caera Calbraton [HZ] Chapter 7 Coputaton of Caera Matrx Soe sldes n ths lecture are courtesy to rofs. J. once, F-F L Slvo Savarese Lecture - 4- Jan- 5
Calbraton rocedure Caera Calbraton oolbox for Matlab J. Bouguet [998-] http://www.vson.caltech.edu/bouguetj/calb_doc/ndex.htlexaples
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