Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH

Pinhole camera model

Pinhole camera model

Principal point offset

Principal point offset calibration matrix

Camera rotation and translation

CCD camera

Finite projective camera 11 dof (5+3+3) non-singular 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 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) Q R =( )-1= -1 -1

Cameras at infinity Camera center at infinity Affine and non-affine cameras Definition: affine camera has P3T=(0,0,0,1)

Affine cameras

Camera calibration

Resectioning

Basic equations

Basic equations n  6 points minimal solution P has 11 dof, 2 independent eq./points 5½ correspondences needed (say 6) Over-determined solution n  6 points minimize subject 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

Geometric error

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 …