1. Camera Models A camera is a mapping between the 3D world and a 2D image The principal camera of interest is central projection

2. Central Projection Cameras modeling Central projection are specialization of the general projective camera. It is examined using the tools of projective geometry. Specialized models fall into two classes: (1) camera with a finite centre. (2) cameras with centre at infinity Affine camera is an important example

3. Finite cameras The basic pin hole model(most specialized)

4. Pinhole camera A point in space with coordinates X =(x,y,z) T is mapped to a point on the image plane. (x,y,z) T ____ ( f x/z, f y/z) T (5.1) The centre of projection is called the camera centre

5. Principal axis and principal point. The line from the camera centre perpendicular to the image plane is called the principal axis and point where it intersects the image plane is called the principal point. The plane parallel to the image plane and passing through the camera centre is called the principal plane

6. Central projection using homogenous coordinates If the world and image points are represented by homogenous vectors, then P = diag( f, f, 1) [I ! 0]

7. Image and camera coordinate systems

8. Principal point offset The expression (5.1) assumes that the origin of the coordinates in the image plane is the principal point. In practice, it may not be as follows. (x,y,z) T ____ ( f x/z + p x, f y/z + p y ) T Where ( p x, p y ) T are the coordinates of the principal point.

9. Principal point offset 2 Now writing

10. Principal point offset 3 Then (5.3) has the concise form x = K [ I ! 0] x cam (5.5) (x, y,z, 1) as X cam as the camera is assumed to be located at the origin of a Euclidean coordinate system with the principal axis of the camera pointing straight down the z-axis. K is called the camera calibration matrix.

11. The Euclidean transformation between the world and camera coordinate frame

12. Camera rotation and translation represents the coordinates of the camera centre in the world coordinate frame. is an inhomogeneous 3-vector in world coordinate frame R is 3x3 rotation axis

13. Camera Rotation and translation Putting (5.5) and (5.6) together leads to

14. Camera Rotation and translation 2 The parameter contained in K are called internal camera parameters The parameter of R and which relate the orientation and position to a world frame are called the external parameters. A more convenient form of the camera matrix is P = K [ R ! t] (5.8)

15. CCD Cameras The CCD camera may have rectangular pixels, where unit distances in x and y directions are m x and m y, then  x 0 = m x p x  and y 0 = m y p y  x = f m x, and  y = f m y

16. Finite projective camera S is the skew parameter A camera is called a finite projective camera. It has 11 degree of freedom, same as a 3x4 matrix defined up to an arbitrary scale

17. General projective Camera

18. The projective camera A general projective camera P maps world point X to image points x according to x = PX

19. Camera centre

20. Camera centre 2

22. Column vectors The columns of P are p i, i = 1, 2, 3, 4 Then p 1, p 2, p 3 are the vanishing points of the world coordinate x, y, z axes respectively. For example: x axis ahs direction D =(1,0,0,0), which is imaged at p 1 = PD The column p 4 is the image of the world origin

23. The three image points defined by the columns p i, i= 1, 2, 3 of the projection matrix are the vanishing points of the directions of the world axes

24. Row vectors

25. Principal plane

26. Axis planes

27. Summary of the properties of a projective camera P=[M ! p 4 ]

28. Summary of the properties of a projective camera 2

29. Principle point

30. Principle axis

31. Principle axis 2

32. Principle axis 3

33. Two of the three planes defined by the rows of the projection matrix

34. Action of a projective camera on points

35. Back projection of points to rays

36. Back projection of points to rays 2

37. Back projection of points to rays 3

38. Depth of points

39. Depth of points 2

40. Linear optics

41. Decomposition of the camera matrix

42. Finding camera orientation and internal parameters

43. Finding camera orientation and internal parameters 2

44. Example 5.2 The camera matrix

45. Euclidean vs Projective space

46. Euclidean and affine interpretation

47. Cameras at infinity

48. Affine cameras

49. Increasing focal length from left to right

50. Affine camera 2

51. Affine camera 3

52. Affine camera 4

53. Focal length increases as the object distance between the camera increases

54. The image remains the same size, but perspective effects diminish

55. Perspective vs weak perspective projection

56. Orthographic projection

57. Orthographic projection 2

58. Scaled orthographic projection

59. Weak perspective projection

60. Affine camera

61. A general affine camera

62. A general affine camera 2

63. More properties of the affine camera

64. General cameras at infinity

65. Pushbroom camera

66. Pushbroom camera 2

67. Pushbroom camera 3

68. Pushbroom camera – mapping of line

69. Line camera

70. Line camera 2

71. Acquisition geometry of a pushbroom camera