1. Stereo March 8, 2007 Suggested Reading: Horn Chapter 13

2. Stereo Vision Stereo vision: ability to infer information on the 3-D structure and distance of a scene from two (or more images) taken from different view points. Two main problems: 1.Find Correspondences 2.Reconstruction (geometry)

3. Simple Stereo System Coordinates origin at the midpoint of the two camera centers.

4. Simple Stereo System

5. Disparity

6. Stereo Imaging

7. Range vs. Disparity

8. Stereo Calibaration

9. More Practical Stereo Model

10. Epipolar Geometry p OLOL OROR Epipoles Epipolar line O R =O L +T P r = R(P l -T) P r : P in the right camera’s frame P l : P in the left camera’s frame

11. Essential Matrix An important problem: Determine the epipolar geometry. That is, the correspondence between a point on one camera and its epipolar line on the other camera. The rigid transformation not important p OLOL OROR Epipoles Epipolar line T

12. Essential Matrix S has rank two. Dividing by Z r, Z l, Essential matrix E is the mapping between points and epipolar lines!

13. Fundamental Matrix Essential matrix uses camera frames. We need matrix that works directly with pixel frames. F: rank 2 and has eight degree of freedom.

14. Eight Point Algorithm To determine F, we need eight corresponding pairs. Each pair of corresponding points give a linear equation (with variable the entries of F).

15. Eight Point Algorithm Let A be the resulting linear system. Find F as the unit vector that minimizes | A F |. 1.SVD of A = USV. F will be the last row of V (if the diagonal entries of S is in descending order). 2.Recall that F has rank two. SVD of F = USV. Let F = US’V where S’ is obtained from S by replacing the smallest singular value of F with 0.

16. Locating Epipole p OLOL OROR Epipoles Epipolar line T Where is the epipole? Epipole is the null vector of F F has rank two. This means that it has one dimensional null space.