1. More single view geometry Describes the images of planes, lines,conics and quadrics under perspective projection and their forward and backward properties

2. Camera properties • Images acquired by the cameras with the same centre are related by a plane projective transformation • Image entities on the plane at infinity, pinf , do not depend on camera position, only on camera rotation and internal parameters, K

3. Camera properties 2 • The image of a point or a line on pinf , depend on both K and camera rotation. • The image of the absolute conic, w , depends only on K; it is unaffected by camera rotation and position. • w = ( KKT )-1

4. Camera properties 2 • w defines the angle between the rays back-projected from image points • Thus camera rotation can be computed from vanishing points independent from camera position. • In turn, K may be computed from the known angle between rays; in particular, K may be computed from vanishing points corresponding to orthogonal scene directions.

5. Perspective image of points on a plane

6. Action of a projective camera on planes

7. Action of a projective camera on lines

8. Action of a projective camera on lines

9. Line projection

10. Action of a projective camera on conics

11. Action of a projective camera on conics 2

12. On conics

13. Images of smooth surfaces

14. Images of smooth surfaces 2

15. Contour generator and apparent contour: for parallel projection

16. Contour generator and apparent contour: for central projection

17. Action of a projective camera on quadrics • Since intersection and tangency are preserved, the contour generator is a (plane) conic. Thus the apparent contour of a general quadric is a conic, so is the contour generator.

18. Result 7.8

19. On quadrics

20. Result 7.9 • The cone with vertex V and tangent to the quadric is the degenerate quadric • QCO = (VT QV) Q – (QV)(QV)T • Note that QCOV = 0, so that V is the vertex of the cone as assumed.

21. The cone rays of a quadric

22. The cone rays with vertex the camera centre

23. Example 7.10

24. The importance of the camera centre

25. The camera centre

26. Moving image plane

27. Moving image plane 2

28. Moving image plane 3

29. Camera rotation

30. Example

31. (a), (b) camera rotates about camera centre. (c) camera rotates about camera centre and translate

32. Synthetic views

33. Synthetic views. (a) Source image(b) Frontal parallel view of corridor floor

34. Synthetic views. (a) Source image(c) Frontal parallel view of corridor wall

35. Planar panoramic mosaicing

36. Three images acquired by a rotating camera may be registered to the frame of the middle one

37. Planar panoramic mosaicing 1

38. Planar panoramic mosaicing 2

39. Planar panoramic mosaicing 3

40. Projective (reduced) notation

41. Moving camera centre

42. Parallax • Consider two 3-space points which has coincident images in the first view( points are on the same ray). If the camera centre is moved (not along that ray), the iamge coincident is lost. This relative displacement of image points is termed Parallax. • An important special case is when all scene points are coplanar. In this case, corresponding image points are related by planar homography even if the camera centre is moved. Vanishing points, which are points on pinf are related by planar homography for any camera motion.

43. Motion parallax

44. Camera calibration and image of the absolute conic

45. The angles between two rays

46. The angle q between two rays

47. Relation between an image line and a scene plane

48. The image of the absolute conic

49. The image of the absolute conic 2

50. The image of the absolute conic 3