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Camera Model & Camera Calibration. Slides are from Marc Pollefeys @ETH. Pinhole camera model. Pinhole camera model. Principal point offset. principal point. Principal point offset. calibration matrix. Camera rotation and translation. CCD camera. non-singular. Finite projective camera.
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Camera Model &Camera Calibration Slides are from Marc Pollefeys @ETH
Principal point offset principal point
Principal point offset calibration matrix
non-singular Finite projective camera 11 dof (5+3+3) decompose P in K,R,C? {finite cameras}={P4x3 | det M≠0} If rank P=3, but rank M<3, then cam at infinity
=( )-1= -1 -1 R R Q Q Camera matrix decomposition Finding the camera center (use SVD to find null-space) Finding the camera orientation and internal parameters (use RQ decomposition ~QR) (if only QR, invert)
Cameras at infinity Camera center at infinity Affine and non-affine cameras Definition: affine camera has P3T=(0,0,0,1)
Basic equations minimal solution P has 11 dof, 2 independent eq./points • 5½ correspondences needed (say 6) Over-determined solution n 6 points minimizesubject to constraint
Degenerate configurations • More complicate than 2D case • Camera and points on a twisted cubic • Points lie on plane or single line passing • through projection center
Data normalization Less obvious (i) Simple, as before (ii) Anisotropic scaling
Gold Standard algorithm • Objective • Given n≥6 2D to 2D point correspondences {Xi↔xi’}, determine the Maximum Likelyhood Estimation of P • Algorithm • Linear solution: • Normalization: • DLT: • Minimization of geometric error: using the linear estimate as a starting point minimize the geometric error: • Denormalization: ~ ~ ~
Calibration example • Canny edge detection • Straight line fitting to the detected edges • Intersecting the lines to obtain the images corners • typically precision <1/10 • (HZ rule of thumb: 5n constraints for n unknowns
Errors in the world Errors in the image and in the world
Radial distortion short and long focal length
Correction of distortion Choice of the distortion function and center • Computing the parameters of the distortion function • Minimize with additional unknowns • Straighten lines • …