1. Epipolar Geometry and the Fundamental Matrix F
The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates this geometry
x F x’ = 0

2. Epipolar geometry
The Epipolar geometry depends only on the internal parameters of the cameras and the relative pose.
A point X in 3 space is imaged in 2 views: x and x’
X, x, x’ and the camera centre C are coplanar in the plane p
The rays back-projected from x and x’ meet at X

3. Point correspondence geometry
Fig. 8.1
Point correspondence geometry

4. Point correspondence geometry

5. Epipolar Geometry
Fig. 8.2

6. Epipolar geometry

7. The geometric entities involved in epipolar geometry

8. Fig 8.3

9. Converging cameras

10. Fig 8.4

11. Motion parallel to the image plane

12. Fig. 8.5
Geometric derivation

13. Point transfer via a plane

14. The fundamental matrix F
x  l’
Geometric Derivation
Step 1: Point transfer via a plane
There is a 2D homography Hp mapping
each xi to xi’
Step 2: Constructing the epipolar line

15. Constructing the epipolar line

16. Cross products
If a = ( a1, a2 , a3)T is a 3-vector, then one define a corresponding skew-sysmmetric matrix as follows:

17. Cross products 2 Matrix [a]x is singular and a is its null vector
a x b = ( a2b3 - a3b2, a3b1 - a1b3 , a1b2 – a2b1)T
a x b = [a]x b =( aT [b]x )T

18. Algebraic derivation

19. Algebraic derivation 2

20. Example 8.2

21. Example 8.2 b

22. Properties of the fundamental matrix (a)

23. Properties of the fundamental matrix (b)

24. Summary of the Properties of the fundamental matrix 1

25. Summary of the properties of the fundamental matrix 2

26. Epipolar line homography 1
Fig. 8.6a
Epipolar line homography 1

27. Epipolar line homography 2
Fig. 8.6 b
Epipolar line homography 2

28. Epipolar line homography

29. The epipolar line homography

30. A pure camera motion

31. Pure translation

32. Fig. 8.8

33. Pure translation motion

34. Example of pure translation

35. Fig. 8.9
General camera motion

36. General camera motion

37. Example of general motion

38. Pure planar motion

39. Retrieving the camera matrices Using F to determine the camera matrices of 2 views
Projective invariance and canonical cameras
Since the relationships l’ = Fx and
x’ F x = 0 are projective relationships
which

40. Projective invariance and canonical cameras
The camera matrix relates 3-space measurements to image measurements and so depends on both the image coordinate frame and the choice of world coordinate frame.
F is unchanged by a projective transformation of 3-space.

41. Projective invariance and canonical cameras 2

42. Canonical form camera matrices

43. Projective ambiguity of cameras given F

44. Projective ambiguity of cameras given F 2

45. Projective ambiguity of cameras given F 3

46. Canonical cameras given F

47. Canonical cameras given F 2

48. Canonical cameras given F 3

49. Canonical cameras given F 4

50. The Essential Matrix

51. Normalized Coordinates

52. Normalized coordinates 2

53. Normalized coordinates 3

54. Properties of the Essential Matrix

55. Result 8.17 on Essential matrix

56. Result 8.17 on Essential matrix 2

57. Extraction of cameras from the Essential Matrix

59. Determine the t part of the camera matrix P’

60. Result 8.19

61. Geometrical interpretation of the four solutions

62. Geometrical interpretation of the four solutions 2

63. The 4 possible solutions for calibrated reconstruction from E