1. Cameras and Projections Dan Witzner Hansen Course web page: www.itu.dk/courses/MCV Email: witzner@itu.dk

2. Previously in Computer Vision…. Homographies Estimating homographies Applications (Image rectification)

3. Outline Projections Pinhole cameras Perspective projection –Camera matrix –Camera calibration matrix Affine Camera Models

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

6. Pinhole camera model

8. Principal point offset principal point

9. Principal point offset calibration matrix

10. Camera rotation and translation

11. CCD camera

12. 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

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

14. 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:

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

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

17. The principal point principal point

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

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) Q R =( ) -1 = -1 -1 Q R (if only QR, invert)

20. Euclidean vs. projective general projective interpretation Meaningful 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

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

22. Affine cameras

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

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

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

26. 1.Affine camera= proj 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)

27. Next: Camera calibration

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

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

30. 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: