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

2. Content Background: Projective geometry (2D, 3D), Parameter estimation, Algorithm evaluation. Single View: Camera model, Calibration, Single View Geometry. Two Views: Epipolar Geometry, 3D reconstruction, Computing F, Computing structure, Plane and homographies. Three Views: Trifocal Tensor, Computing T. More Views: N-Linearities, Multiple view reconstruction, Bundle adjustment, auto- calibration, Dynamic SfM, Cheirality, Duality

3. Multiple View Geometry course schedule (subject to change) Jan. 7, 9Intro & motivationProjective 2D Geometry Jan. 14, 16(no class)Projective 2D Geometry Jan. 21, 23Projective 3D Geometry(no class) Jan. 28, 30Parameter Estimation Feb. 4, 6Algorithm EvaluationCamera Models Feb. 11, 13Camera CalibrationSingle View Geometry Feb. 18, 20Epipolar Geometry3D reconstruction Feb. 25, 27Fund. Matrix Comp.Structure Comp. Mar. 4, 6Planes & HomographiesTrifocal Tensor Mar. 18, 20Three View ReconstructionMultiple View Geometry Mar. 25, 27MultipleView ReconstructionBundle adjustment Apr. 1, 3Auto-CalibrationPapers Apr. 8, 10Dynamic SfMPapers Apr. 15, 17CheiralityPapers Apr. 22, 24DualityProject Demos

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

5. 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: ~~ ~

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

10. 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)

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

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

13. 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

14. 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

15. The importance of the camera center

16. Moving the image plane (zooming)

17. Camera rotation conjugate rotation

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

19. Planar homography mosaicing

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

21. Planar homography mosaicing more examples

22. Projective (reduced) notation

23. Moving the camera center motion parallax epipolar line

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

25. 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

26. 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)

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

28. The calibrating conic

29. Vanishing points

30. Vanishing lines

32. Orthogonality relation

33. Calibration from vanishing points and lines

35. Next class: Two-view geometry Epipolar geometry