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Camera Model & Camera Calibration

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Camera Model & Camera Calibration

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

  2. Pinhole camera model

  3. Pinhole camera model

  4. Principal point offset principal point

  5. Principal point offset calibration matrix

  6. Camera rotation and translation

  7. CCD camera

  8. 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

  9. =( )-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)

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

  11. Affine cameras

  12. Camera calibration

  13. Resectioning

  14. Basic equations

  15. 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

  16. 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

  17. Data normalization Less obvious (i) Simple, as before (ii) Anisotropic scaling

  18. Geometric error

  19. 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: ~ ~ ~

  20. 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

  21. Errors in the world Errors in the image and in the world

  22. Radial distortion short and long focal length

  23. Correction of distortion Choice of the distortion function and center • Computing the parameters of the distortion function • Minimize with additional unknowns • Straighten lines • …