Computational Vision. Camera Calibration

Transcription

1 Comutatonal Vson Camea Calbaton uo hate 6 Camea Calbaton Poblem: Estmate amea s etns & ntns aametes MthdU Method: Use mage(s) () o knon sene ools: Geomet amea models SVD and onstaned least-squaes Lne etaton methods

2 Coodnate Fames Camea Coodnate Fame Pel Coodnates Intns Paametes Etns Paametes Wold Coodnate Fame Image Coodnate Fame Wh Calbate? Image ont Sene a Image Calbaton: elates onts n the mage to as n the sene

3 Wh Calbate? Image ont Sene a Image Calbaton: elates onts n the mage to as n the sene Pesete Camea Cente o Pojeton () () =() =() /= / : eete oal length: dstane o mage lane om O = * / = * / = 3

4 Etns Paametes P=R(P-) anslaton olloed b otaton Etns Paametes ( nd omulaton) R same as beoe P=R P + deent Rotaton olloed b tanslaton 4

5 he Rotaton Mat R * R = R * R = I => - R = R Othonomal Mat Degees o eedom? I= Intns Paametes 5

6 Image and Camea Fames amea amea (m m) amea mage (oo) mage Geomet Model = = 3D Pont n Camea Coodnate Fame ansomaton om Image to Camea Fame (ooss) o dstoton! ansomaton om Wold to Camea Fame Pesete ojeton ( R ) Pont n Camea Fame 6

7 Camea Calbaton: Issues Whh aametes need to be estmated Foal length mage ente aset ato Radal dstotons What knd o aua s needed laton deendent What knd o albaton objet s used One lane man lanes Comlated thee dmensonal objet Camea Calbaton Calbaton objet Etated eatues 7

8 8 Camea Calbaton Etat entes o les Bas Equatons m m o o s s

9 9 Bas Equatons o o Bas Equatons o o

10 Etns Paametes ) Rotaton mat R (33) ) anslaton eto (3) Bas Equatons Intns Paametes ) =/s length n eete hoontal el se unts ) α=s/s aset ato 3) (oo) mage ente oodnates 4) Radal dstoton oeents otal numbe o aametes (eludng dstoton): 8 Bas Equatons o o ) ssume that mage ente s knon ) Sole o the emanng aametes 3) Use mage onts ( ) and the oesondng old onts [ ]

11 Bas Equatons () ) ssume that mage ente s knon ) Sole o the emanng aametes 3) Use mage onts ( ) and the oesondng old onts [ ] Bas Equatons ( ( 3 3 ) ) () ) ssume that mage ente s knon ) Sole o the emanng aametes 3) Use mage onts ( ) and the oesondng old onts [ ]

12 Bas Equatons (3) Bas Equatons (3) Ho ould e sole ths sstem?

13 3 Bas Equatons (3) Ho ould e sole ths sstem? Rank o mat? Soluton u to a sale ato Sngula Value Deomoston UDV end 6 : m n : m n U: m m olumns othogonal unt etos V: n n -//- D: m n dagonal he dagonal elements σ ae the sngula alues σ>= σ>= >= σn >=

14 Sngula Value Deomoston end 6 UDV Squae non-sngula σ!= Fo squae C=σ/σ s the ondton numbe 3 Fo etangula # o non-eo σ s the ank 4 Fo squae non-sngula : VD U 5 Fo squae seudonese: VD U 6 Sngula alues o = squae oots o egenalues o and 7 Columns o U V Egenetos o 8 Fobenus nom o a mat Sngula Value Deomoston end 6 UDV I ank()=n- (7 n ou ase) then the soluton s the egeneto hh oesonds to the OL eo egenalue Soluton u to a sale ato 4

15 5 Solng o (3) Ho ould e sole ths sstem: SVD Soluton: Uknon sale ato γ=? set ato α=? ) ( 3 3 Solng o and?

16 6 Solng o and? ) ( ) ( b Ho ould e sole ths sstem? Solng o and? ) ( ) ( b Ho ould e sole ths sstem? ^ b ) ^ ( Soluton n the least squaes sense

17 Camea Cente Camea Models (lnea esons) Elegant deomoston o dstoton! Homogeneous Coodnates Measued Pel (m m)? Wold Pont ( ) 7

18 Camea Calbaton Othe method u P u Etated eatues Ste : Estmate P Ste : Deomose P nto ntenal and etenal aametes RC Camea Calbaton: Ste u P u u Etated eatues u Eah ont () ges us to equatons 8

19 9 Camea Calbaton: Ste Etated eatues Eah one () ges us to equatons Camea Calbaton: Ste n Etated eatues n onts ges us n equatons

20 Camea Calbaton: Ste n Etated eatues mn We need to sole In the esene o nose e need to sole he soluton s gen b the egeneto th the smallest egenalue o Camea Calbaton: Ste he esult an be moed though non-lnea mnmaton u mn u Etated eatues P

21 Camea Calbaton: Ste he esult an be moed though non-lnea mnmaton u mn u Etated eatues P Mnme the dstane beteen the edted and deteted eatues