1. Projective Geometry

6. Projection

7. Vanishing lines m and n

8. Projective Plane (Extended Plane)

9. Projective Plane How??? Ordinary plane

10. Point Representation A point in the projective plane is represented as a ray in R 3

11. Projective Geometry

12. Homogeneous coordinates Homogeneous representation of 2D points and lines equivalence class of vectors, any vector is representative Set of all equivalence classes in R 3  (0,0,0) T forms P 2 The point x lies on the line l if and only if Homogeneous coordinates Inhomogeneous coordinates but only 2DOF Note that scale is unimportant for incidence relation

13. Projective Geometry

14. Projective plane = S 2 with antipodal points identified Ordinary plane is unbound Projective plane is bound!

15. Projective Geometry

17. Pappus’ Theorem

20. Conic Section

28. Form of Conics

29. Transformation Projective: incidence, tangency Affine: plane at infinity, parallelism Similarity: absolute conics

30. Circular Point Circular points

31. Euclidean Transformation Any transformation of the projective plane which leaves the circular points fixed is a Euclidean transformation, and Any Euclidean transformation leaves the circular points fixed. A Euclidean transformation is of the form:

32. Euclidean Transformation

33. Calibration

34. Use circular point as a ruler…

35. Calibration

36. Today Cross ratio More on circular points and absolute conics Camera model and Zhang’s calibration Another calibration method

37. Transformation Let X and X’ be written in homogeneous coordinates, when X’=PX P is a projective transformation when….. P is an affine transformation when….. P is a similarity transformation when…..

38. Transformation Projective Affine Similarity Euclidean

39. Matrix Representation

40. Invariance Mathematician loves invariance ! Fixed point theorem Eigenvector

41. Cross Ratio Projective line P = (X,1) t Consider

42. Cross Ratio

43. Consider determinants: Rewritting So we have Consider

44. Cross Ratio How do we eliminate |T| and the coefficients The idea is to use the ratio. Consider and The remaining coefficients can be eliminated by using the fourth point

45. Pinhole Camera

46. 3x4 projection matrix 3x3 intrinsic matrix Extrinsic matrix Principle point Skew factor

47. Pinhole Camera

48. Absolute Conic

50. Important: absolute conic is invariant to any rigid transformation We can writeand That is, and obtain

51. Absolute Conic Now consider the image of the absolute conic It is defined by

52. Typical Calibration 1.Estimate the camera projection matrix from correspondence between scene points and image points (Zhang p.12) 2.Recover intrinsic and extrinsic parameters

53. Typical Calibration P[3][4], B[3][3], b[3]

54. Calibration with IAC Can we calibrate without correspondence? (British Machine Vision)

55. Calibration with IAC

56. From Zhang’s, the image of the absolute conic is the conic Let’s assume that the model plane is on the X-Y plane of the world coordinate system, so we have:

57. Calibration with IAC Points on the model plane with t=0 form the line at infinity It is sufficient to consider model plane in homogeneous coordinates We know that the circular points I = (1,i,0,0) T and J = (1,-i,0,0) T must satisfy Let the image of I and J be denoted by

58. Calibration with IAC Consider the circle in the model plane with center (Ox,Oy,1) and radius r. This circle intersects the line at infinitywhen or Any circle (any center, any radius) intersects line at infinity in the two circular points The image of the circle should intersect the image of the line at infinity (vanishing line) in the image of the two circular points

59. Calibration with IAC