Calbratng a sngle camera Odlon Redon, Cclops, 94
Our goal: Recover o 3D structure Recover o structure rom one mage s nherentl ambguous???
Sngle-vew ambgut
Sngle-vew ambgut Rashad Alakbarov shadow sculptures
Sngle-vew ambgut Ames room
Our goal: Recover o 3D structure We wll need mult-vew geometr
Revew: Pnhole camera model world coordnate sstem Normalzed (camera) coordnate sstem: camera center s at the orgn, the prncpal as s the z-as, and aes o the mage plane are parallel to and aes o the world Goal o camera calbraton: go rom world coordnate sstem to mage coordnate sstem
) /, / ( ),, ( Z Y Z Z Y! Z Y Z Y Z Y! Revew: Pnhole camera model P
Prncpal pont p p Prncpal pont (p): pont where prncpal as ntersects the mage plane Normalzed coordnate sstem: orgn o the mage s at the prncpal pont Image coordnate sstem: orgn s n the corner
) /, / ( ),, ( p Z Y p Z Z Y + +! + + Z Z p Y Z p Z Y! Prncpal pont oset We want the prncpal pont to map to (p, p ) nstead o (,) p p Z Y p p
+ + Z Y p p Z Zp Y Zp Prncpal pont oset p p K calbraton matr [ ] P K I prncpal pont: ), ( p p p p
p p m m K β α β α Pel coordnates m pels per meter n horzontal drecton, m pels per meter n vertcal drecton Pel sze: m m pels/m m pels
Camera rotaton and translaton In general, the camera coordnate rame wll be related to the world coordnate rame b a rotaton and a translaton Converson rom world to camera coordnate sstem (n non-homogeneous coordnates): ~ cam ( ) R ~ C ~ coords. o pont n camera rame coords. o a pont n world rame coords. o camera center n world rame
Camera rotaton and translaton ~ cam ( ) R ~ C ~ cam ~ cam R RC ~ ~ R RC ~ [ ] K[ R ] K I RC ~ cam P K[ R t], t RC ~
Camera parameters Intrnsc parameters Prncpal pont coordnates Focal length Pel magncaton actors Skew (non-rectangular pels) Radal dstorton p p m m β α β α K [ ] t P K R
Camera parameters P K[ R t] Intrnsc parameters Prncpal pont coordnates Focal length Pel magncaton actors Skew (non-rectangular pels) Radal dstorton Etrnsc parameters Rotaton and translaton relatve to world coordnate sstem P [ ~ ] K R RC What s the projecton o the camera center? PC ~ ~ C [ RC] K R coords. o camera center n world rame he camera center s the null space o the projecton matr!
Camera calbraton Z Y λ λ λ [ ] t K R Source: D. Hoem
Camera calbraton Gven n ponts wth known 3D coordnates and known mage projectons, estmate the camera parameters P?
P λ Camera calbraton: Lnear method P 3 P P P 3 P P P wo lnearl ndependent equatons
Camera calbraton: Lnear method P has degrees o reedom One D/3D correspondence gves us two lnearl ndependent equatons 6 correspondences needed or a mnmal soluton Homogeneous least squares: nd p mnmzng Ap Soluton gven b egenvector o A A wth smallest egenvalue p A 3 P P P n n n n n n!!!
Camera calbraton: Lnear method Note: or coplanar ponts that sats Π, we wll get degenerate solutons (Π,,), (,Π,), or (,,Π) Ap 3 P P P n n n n n n!!!
Camera calbraton: Lnear method he lnear method onl estmates the entres o the projecton matr: What we ultmatel want s a decomposton o ths matr nto the ntrnsc and etrnsc parameters: State-o-the-art methods use nonlnear optmzaton to solve or the parameter values drectl [ ] t K R Z Y λ λ λ
Camera calbraton: Lnear method Advantages: eas to ormulate and solve Dsadvantages Doesn t drectl tell ou camera parameters Doesn t model radal dstorton Can t mpose constrants, such as known ocal length and orthogonalt Non-lnear methods are preerred Dene error as sum o squared dstances between measured D ponts and estmated projectons o 3D ponts Mnmze error usng Newton s method or other non-lnear optmzaton Source: D. Hoem
A taste o mult-vew geometr: rangulaton Gven projectons o a 3D pont n two or more mages (wth known camera matrces), nd the coordnates o the pont
rangulaton Gven projectons o a 3D pont n two or more mages (wth known camera matrces), nd the coordnates o the pont? O O
rangulaton We want to ntersect the two vsual ras correspondng to and, but because o nose and numercal errors, the don t meet eactl? O O
rangulaton: Geometrc approach Fnd shortest segment connectng the two vewng ras and let be the mdpont o that segment O O
rangulaton: Nonlnear approach Fnd that mnmzes d (,P ) + d (,P )? P P O O
rangulaton: Lnear approach b a b a ] [ z z z b b b a a a a a a P P λ λ P P ]P [ ]P [ Cross product as matr multplcaton:
rangulaton: Lnear approach λ λ P P P P [ [ ]P ]P wo ndependent equatons each n terms o three unknown entres o