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Projective Geometry

Projective Geometry. Projective Geometry. Projective Geometry. Projective Geometry. Projective Geometry. Projection. Projection. Vanishing lines m and n. Projective Plane (Extended Plane). Projective Plane. Ordinary plane. How???. Point Representation.

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Projective Geometry

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  1. Projective Geometry

  2. Projective Geometry

  3. Projective Geometry

  4. Projective Geometry

  5. Projective Geometry

  6. Projection

  7. Projection Vanishing lines m and n

  8. Projective Plane (Extended Plane)

  9. Projective Plane Ordinary plane How???

  10. Point Representation A point in the projective plane is represented as a ray in R3

  11. Projective Geometry

  12. Homogeneous coordinates but only 2DOF Inhomogeneous coordinates Homogeneous coordinates Homogeneous representation of 2D points and lines The point x lies on the line l if and only if Note that scale is unimportant for incidence relation equivalence class of vectors, any vector is representative Set of all equivalence classes in R3(0,0,0)T forms P2

  13. Projective Geometry

  14. Projective Geometry Projective plane = S2 with antipodal points identified Ordinary plane is unbound Projective plane is bound!

  15. Projective Geometry

  16. Projective Geometry

  17. Pappus’ Theorem

  18. Pappus’ Theorem

  19. Pappus’ Theorem

  20. Conic Section

  21. Conic Section

  22. Conic Section

  23. Conic Section

  24. Conic Section

  25. Conic Section

  26. Conic Section

  27. 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. Calibration 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. Cross Ratio 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. Pinhole Camera Skew factor Principle point Extrinsic matrix 3x3 intrinsic matrix 3x4 projection matrix

  47. Pinhole Camera

  48. Absolute Conic

  49. Absolute Conic

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

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