More on single-view geometry class 10

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

2. Multiple View Geometry course schedule(subject to change)

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

4. 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: ~ ~ ~

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

9. 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)

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

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

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

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

14. The importance of the camera center

15. Moving the image plane (zooming)

16. Camera rotation conjugate rotation

17. Synthetic view • Compute the homography that warps some a rectangle to the correct aspect ratio • warp the image

18. Planar homographymosaicing

19. close-up: interlacing can be important problem!

20. Planar homography mosaicing more examples

21. Projective (reduced) notation

22. Moving the camera center motion parallax epipolar line

23. What does calibration give? An image l defines a plane through the camera center with normal n=KTl measured in the camera’s Euclidean frame

24. The image of the absolute conic mapping between p∞ to an image is given by the planar homogaphy x=Hd, with H=KR image of the absolute conic (IAC) • IAC depends only on intrinsics • angle between two rays • DIAC=w*=KKT • w  K (choleskyfactorisation) • image of circular points

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

26. Orthogonality = pole-polar w.r.t. IAC

27. The calibrating conic

28. Vanishing points

29. ML estimate of a vanishing point from imaged parallel scene lines

30. Vanishing lines

33. Orthogonality relation

38. Five constraints gives us five equations and can determine w

39. Calibration from vanishing points and lines Principal point is the orthocenter of the trinagle made of 3 orthogonol vanishing lines Assumes zero skew, square pixels and 3 orthogonal vanishing points

40. Assume zero skew, square pixels,  calibrating conic is a circle; How to find it, so that we can get K?

41. Assume zero skew , square pixels, and principal point is at the image center Then IAC is diagonal{1/f^2, 1/f^2,1) i.e. one degree of freedom need one more Constraint to determine f, the focal length  two vanishing points corresponding To orthogonal directions.

44. Next class: Two-view geometryEpipolar geometry