1. Camera Model & Camera Calibration
Slides are from Marc

2. Pinhole camera model

3. Pinhole camera model

4. Principal point offset

5. Principal point offset
calibration matrix

6. Camera rotation and translation

7. CCD camera

8. 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

9. =( )-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=

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 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

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

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