1. Scene planes and homographies

2. Homographies given the plane and vice versa

3. Proof of result 12.1

4. Example 12.2 A calibrated stereo rig

5. A calibrated stereo rig 2

6. A calibrated stereo rig 3

7. The homography induced by a plane Fig.12.1

8. Fig 12.1 Legend

9. Homographies compatible with epipolar Geometry

10. Two sets of 4 arbitrary points from 2 images

11. Epipolar geometry define conditions on homographies

12. Counting degrees of freedom

13. Compatibility constraints Fig.12.2 a e’ = H e

14. Compatibility constraints 2 Fig. 12.2 b H T l e ’ = l e

15. Compatibility constraints 3 Fig. 12.2 c

16. Fig 12.2 Compatibility constraints

17. Result 12.3

18. Homographies are compatible with fundamental matrix

19. Corollary 12.4

20. Result 12.5

21. 13.6 Plane induced homographies given F and image correspondences: (a) 3 points, (b) a line and a point

22. 12.2.1 Three points

23. Three points

24. The first (explicit) method is preferred

25. Degenerate geometry for an implicit computation of the homography Fig. 12.3

26. Fig. 12.3 Legend

27. Determining the points X i is not necessary in first method All that is important

28. Result 12.6

29. Proof

30. Proof 2

31. Consistency conditions

32. Consistency conditions 2

33. Algorithms 12.1 The optimal estimate of homography induced by a plane defined by 3 points

34. 12.2.2 A point and line

35. A one parameter family of homographies Fig 12.4 (a), (b)

36. Fig 12.4 Legend

37. Result 12.7

38. Proof of result 12.7

39. Proof of result 12.7 (2)

40. Proof of result 12.7 (3)

41. Result 12.8

42. Result 12.8 2

43. Geometric interpretation of the point map H(  Explore the

44. A homography between corresponding line images Fig. 12.5

45. Fig. 12.5 Legend

46. Degenerate homographies

47. Degenerate homographies 2

48. A degenerate homography Fig. 12.6

49. Fig. 12.6 Legend

50. 12.3 Computing F given the homography induced by a plane

51. Plane induced parallax

52. Plane induced parallax Fig. 12.7

53. Fig. 12.7 Legend

54. Plane induced parallax 2 Fig. 12.8

55. Fig. 12.8 Legend

56. Plane induced parallax 2

57. Algorithm 12.2 Computing F given the correspondence of 6 points, 4 of which are coplanar

58. Fundamental matrix from 6 points of which 4 are coplanar Fig. 12.9

59. Fig. 12.9 Legend

60. Projective Depth

61. Example 12.9

62. Binary space partition: left and right images Fig. 12.10 a,b

63. (c ) Points with known correspondence (d) A triplet of points selected from ( c ) and this triplet defines a plane Fig. 12.10 c,d

64. (e) Points on one side of the plane (f) Points on the other side Fig. 12.10 e, f

65. Fig 12.10 Legend

66. Two planes

67. Two planes 2

68. The action of the map H = H 2 -1 H 1 on x Fig. 12.11

69. Fig. 12.11 Legend

70. Two planes 3 Up to this points, the results of this chapter have been entirely projective

71. 12.4 The infinite homography H inf

72. The infinite homography H inf 2

73. The infinite homography H inf 3

74. Vanishing points and lines

75. The infinite homography H inf maps vanishing points between images Fig. 12.12

76. Affine and metric reconstruction

77. Affine and metric reconstruction 2

78. Affine and metric reconstruction 3

79. Stereo Correspondence

80. Reducing the search region using H inf Fig 12.13

81. Fig. 12.13 Legend