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Cameras and Projections

Cameras and Projections. Dan Witzner Hansen Course web page: www.itu.dk/courses/MCV Email: witzner@itu.dk. Previously in Computer Vision…. Homographies Estimating homographies Applications (Image rectification). Outline. Projections Pinhole cameras Perspective projection

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Cameras and Projections

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  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.

  5. Pinhole camera model

  6. Pinhole camera model

  7. Principal point offset principal point

  8. Principal point offset calibration matrix

  9. Camera rotation and translation

  10. CCD camera

  11. 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

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

  13. 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:

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

  15. Row vectors note: p1,p2 dependent on image reparametrization

  16. principal point The principal point

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

  18. =( )-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)

  19. 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

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

  21. Affine cameras

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

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

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

  25. A hierarchy of affine cameras Affine camera (8dof) • Affine camera= proj camera with principal plane coinciding with P∞ • Affine camera maps parallel lines to parallel lines • No center of projection, but direction of projection PAD=0 • (point on P∞)

  26. Next: Camera calibration

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

  28. Depth of points (PC=0) (dot product) If , then m3 unit vector in positive direction

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

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