1. Camera Models class 8 Multiple View Geometry Comp 290-089 Marc Pollefeys

2. Backward propagation of covariance X f -1 P X  Over-parameterization J f v Forward propagation of covariance Monte-Carlo estimation of covariance

3. Single view geometry Camera model Camera calibration Single view geom.

4. Pinhole camera model

6. Principal point offset principal point

7. Principal point offset calibration matrix

8. Camera rotation and translation

9. CCD camera

10. Finite projective camera non-singular 11 dof (5+3+3) decompose P in K,R,C? {finite cameras}={P 4x3 | det M≠0} If rank P=3, but rank M<3, then cam at infinity

11. Camera anatomy Camera center Column points Principal plane Axis plane Principal point Principal ray

12. Camera center null-space camera projection matrix For all A all points on AC project on image of A, therefore C is camera center Image of camera center is (0,0,0) T, i.e. undefined Finite cameras: Infinite cameras:

13. Column vectors Image points corresponding to X,Y,Z directions and origin

14. Row vectors note: p 1,p 2 dependent on image reparametrization

15. The principal point principal point

16. The principal axis vector vector defining front side of camera (direction unaffected) because

17. Action of projective camera on point Forward projection Back-projection (pseudo-inverse)

18. Depth of points (dot product) (PC=0) If, then m 3 unit vector in positive direction

19. Camera matrix decomposition Finding the camera center (use SVD to find null-space) Finding the camera orientation and internal parameters (use RQ decomposition ~QR)

20. When is skew non-zero? 1  arctan(1/s) for CCD/CMOS, always s=0 Image from image, s≠0 possible (non coinciding principal axis) resulting camera:

21. Euclidean vs. projective general projective interpretation Meaningfull decomposition in K,R,t requires Euclidean image and space Camera center is still valid in projective space Principal plane requires affine image and space Principal ray requires affine image and Euclidean space

22. Cameras at infinity Camera center at infinity Affine and non-affine cameras Definition: affine camera has P 3T =(0,0,0,1)

23. Affine cameras

24. modifying p 34 corresponds to moving along principal ray

25. Affine cameras now adjust zoom to compensate

26. Error in employing affine cameras point on plane parallel with principal plane and through origin, then general points

27. Affine imaging conditions Approximation should only cause small error  much smaller than d 0 2.Points close to principal point (i.e. small field of view)

28. Decomposition of P ∞ absorb d 0 in K 2x2 alternatives, because 8dof (3+3+2), not more

29. Summary parallel projection canonical representation calibration matrix principal point is not defined

30. A hierarchy of affine cameras Orthographic projection Scaled orthographic projection (5dof) (6dof)

31. A hierarchy of affine cameras Weak perspective projection (7dof)

32. 1.Affine camera=camera with principal plane coinciding with  ∞ 2.Affine camera maps parallel lines to parallel lines 3.No center of projection, but direction of projection P A D=0 (point on  ∞ ) A hierarchy of affine cameras Affine camera (8dof)

33. Pushbroom cameras Straight lines are not mapped to straight lines! (otherwise it would be a projective camera) (11dof)

34. Line cameras (5dof) Null-space PC=0 yields camera center Also decomposition

35. Next class: Camera calibration