1. More on single-view geometry class 10 Multiple View Geometry Comp 290-089 Marc Pollefeys

2. Gold Standard algorithm Objective Given n≥6 2D to 2D point correspondences {X i ↔x i ’}, determine the Maximum Likelyhood Estimation of P Algorithm (i)Linear solution: (a)Normalization: (b)DLT (ii)Minimization of geometric error: using the linear estimate as a starting point minimize the geometric error: (iii)Denormalization: ~~ ~

6. More Single-View Geometry Projective cameras and planes, lines, conics and quadrics. Camera calibration and vanishing points, calibrating conic and the IAC

7. Action of projective camera on planes The most general transformation that can occur between a scene plane and an image plane under perspective imaging is a plane projective transformation (affine camera-affine transformation)

8. Action of projective camera on lines forward projection back-projection

9. Action of projective camera on conics back-projection to cone example:

10. Images of smooth surfaces The contour generator  is the set of points X on S at which rays are tangent to the surface. The corresponding apparent contour  is the set of points x which are the image of X, i.e.  is the image of  The contour generator  depends only on position of projection center,  depends also on rest of P

11. Action of projective camera on quadrics back-projection to cone The plane of  for a quadric Q is camera center C is given by  =QC (follows from pole-polar relation) The cone with vertex V and tangent to the quadric Q is

12. The importance of the camera center

13. Moving the image plane (zooming)

14. Camera rotation conjugate rotation

15. Synthetic view (i)Compute the homography that warps some a rectangle to the correct aspect ratio (ii)warp the image

16. Planar homography mosaicing

17. more examples

18. Projective (reduced) notation

19. Moving the camera center motion parallax epipolar line

20. What does calibration give? An image line l defines a plane through the camera center with normal n=K T l measured in the camera’s Euclidean frame

21. The image of the absolute conic mapping between  ∞ to an image is given by the planar homogaphy x=Hd, with H=KR image of the absolute conic (IAC) (i)IAC depends only on intrinsics (ii)angle between two rays (iii)DIAC=  * =KK T (iv)   K (cholesky factorisation) (v)image of circular points

22. A simple calibration device (i)compute H for each square (corners  (0,0),(1,0),(0,1),(1,1)) (ii)compute the imaged circular points H(1,±i,0) T (iii)fit a conic to 6 circular points (iv)compute K from  through cholesky factorization (= Zhang’s calibration method)

23. Orthogonality =conjuacy and pole-polar w.r.t. IAC

24. The calibrating conic

25. Vanishing points

26. Vanishing lines

28. Orthogonality relation

29. Calibration from vanishing points and lines

31. Next class: Two-view geometry Epipolar geometry