1. Multiple View Geometry

2. THE GEOMETRY OF MULTIPLE VIEWS Reading: Chapter 10. Epipolar Geometry The Essential Matrix The Fundamental Matrix The Trifocal Tensor The Quadrifocal Tensor

3. Epipolar Geometry Epipolar Plane Epipoles Epipolar Lines Baseline

4. Epipolar Constraint Potential matches for p have to lie on the corresponding epipolar line l’. Potential matches for p’ have to lie on the corresponding epipolar line l.

5. Epipolar Constraint: Calibrated Case Essential Matrix (Longuet-Higgins, 1981)

6. Properties of the Essential Matrix E p’ is the epipolar line associated with p’. E T p is the epipolar line associated with p. E e’=0 and E T e=0. E is singular. E has two equal non-zero singular values (Huang and Faugeras, 1989). T T

7. Epipolar Constraint: Small Motions To First-Order: Pure translation: Focus of Expansion

8. Epipolar Constraint: Uncalibrated Case Fundamental Matrix (Faugeras and Luong, 1992)

9. Properties of the Fundamental Matrix F p’ is the epipolar line associated with p’. F T p is the epipolar line associated with p. F e’=0 and F T e=0. F is singular. T T

10. The Eight-Point Algorithm (Longuet-Higgins, 1981) | F | =1. Minimize: under the constraint 2

11. Non-Linear Least-Squares Approach (Luong et al., 1993) Minimize with respect to the coefficients of F, using an appropriate rank-2 parameterization.

12. Problem with eight-point algorithm linear least-squares: unit norm vector F yielding smallest residual What happens when there is noise?

13. The Normalized Eight-Point Algorithm (Hartley, 1995) Center the image data at the origin, and scale it so the mean squared distance between the origin and the data points is sqrt(2) pixels: q = T p, q’ = T’ p’. Use the eight-point algorithm to compute F from the points q and q’. Enforce the rank-2 constraint. Output T F T’. T iiii ii

14. Weak-calibration Experiments

15. Epipolar geometry example

16. Example: converging cameras courtesy of Andrew Zisserman