More single view geometry

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# More single view geometry

## More single view geometry

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##### Presentation Transcript

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. 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.

16. Result 7.8

18. 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.

19. The cone rays of a quadric

20. The cone rays with vertex the camera centre

21. Example 7.10

22. The importance of the camera centre

23. The camera centre

24. Moving image plane

25. Moving image plane 2

26. Moving image plane 3

27. Camera rotation

28. Example

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

30. Synthetic views

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

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

33. Planar panoramic mosaicing

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

35. Planar panoramic mosaicing 1

36. Planar panoramic mosaicing 2

37. Planar panoramic mosaicing 3

38. Projective (reduced) notation

39. Moving camera centre

40. 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.

41. Motion parallax

42. Camera calibration and image of the absolute conic

43. The angles between two rays

44. The angle q between two rays

45. Relation between an image line and a scene plane

46. The image of the absolute conic

47. The image of the absolute conic 2

48. The image of the absolute conic 3