1. Geometry 2: A taste of projective geometry Introduction to Computer Vision Ronen Basri Weizmann Institute of Science

2. Summery of last lecture

3. Material covered Pinhole camera model, perspective projection Two view geometry, general case: Epipolar geometry, the essential matrix Camera calibration, the fundamental matrix Two view geometry, degenerate cases Homography (planes, camera rotation) A taste of projective geometry Stereo vision: 3D reconstruction from two views Multi-view geometry, reconstruction through factorization

4. Camera matrix

5. The uncalibrated case: the Fundamental matrix

6. The Fundamental matrix

7. Geometry Geometry – Greek: earth measurement Geometry concerns with shape, size, relative positions, and properties of spaces Euclidean geometry: Point, line, plane Incidence Continuity Order, “between” Parallelism Congruence = invariance: angles, lengths, areas are preserved under rigid transformations

8. Projective geometry How does a plane looks after projection? How does perspective distorts geometry?

9. Plane perspective Pencil of rays

10. Plane perspective Pencil of rays

11. Projective transformation

12. How these change from Eucleadian geometry? Point, line, plane Incidence Continuity Order, “between” Parallelism Congruence Under projective transformation A (straight) line transforms to a line and a conic to a conic But order and parallelism are not preserved Likewise, angles, lengths and areas are not preserved

13. Projective coordinates

14. Projective line

15. Intersection and incidence

16. Ideal points

17. Line at infinity

19. Homography

21. Hierarchy of transformations Rigid Preserves angles, lengths, area, parallelism SimilarityPreserves angles, parallelism AffinePreserves parallelism Homography Preserves cross ratio

22. Camera rotation

23. Planar scene

24. Summary HomographyPerspective (calibrated) Perspective (uncalibrated) Orthographic Form PropertiesOne-to-one (group) Concentric epipolar lines Parallel epipolar lines DOFs 8(5) 8(7)4 Eqs/pnt 2111 Minimal configuration 45+ (8,linear)7+ (8,linear)4 DepthNoYes, up to scale Yes, projective structure Affine structure (third view required for Euclidean structure)

25. Recovering epipolar constraints

27. Interest points (Harris)

28. Descriptor: SIFT ( Scale invariant feature transform)

29. SIFT matches

30. RANSAC

31. Epipolar lines