1. Stanford CS223B Computer Vision, Winter 2006 Lecture 5 Stereo I
Professor Sebastian Thrun
CAs: Dan Maynes-Aminzade, Mitul Saha, Greg Corrado

2. Homework #1

3. Vocabulary Quiz Baseline Epipole Fundamental Matrix Essential Matrix
Stereo Rectification

4. Stereo Vision: Illustration

5. Stereo Example (Stanley Robot)
Disparity map

6. Stereo Example

7. Stereo Vision: Outline
Basic Equations
Epipolar Geometry
Image Rectification
Reconstruction
Correspondence
Dense and Layered Stereo
(Active Range Imaging Techniques)

8. The Two Problems of Stereo
Correspondence (Wed)
Reconstruction (Today)

9. Pinhole Camera Model
Image
plane
Focal length f
Center of
projection

10. Pinhole Camera Model
Image
plane

11. Pinhole Camera Model
Image
plane

12. Basic Stereo Derivations

13. Basic Stereo Derivations

14. What If…?

15. Epipolar Geometry P Pl Pr Yr p p l r Yl Zl Zr Xl fl fr Ol Or Xr

16. Epipolar Geometry r P Pl Pr Epipolar Plane Epipolar Lines p p l Ol eler Or
Epipoles

17. Epipolar Geometry
Epipolar plane: plane going through point P and the centers of projection (COPs) of the two cameras
Epipoles: The image in one camera of the COP of the other
Epipolar Constraint: Corresponding points must lie on epipolar lines

18. Essential Matrix Coordinate Transformation: Coplanarity T, Pl, Pl-T:Pr p p l r Ol el er Or
Coordinate Transformation:
Coplanarity T, Pl, Pl-T:
Resolves to
Essential Matrix

19. Essential Matrix Projective Line: Essential Matrix r P Pl Pr p p l Oler Or
Projective Line:
Essential Matrix

20. Fundamental Matrix
Same as Essential Matrix in Camera Pixel Coordinates
Pixel coordinates
Intrinsic parameters

21. Intrinsic Parameters (See Chapter 2)

22. Computing F: The Eight-Point Algorithm
Problem: Recover F (3-3 matrix of rank 2)
Ides: Get 8 points:
Minimize:
Notice: Argument linear in coefficients of F

23. Computing F: The Eight-Point Algorithm
Run Singular Value Decomposition of A
Appendix A.6, page
See also G. Strang: Linear algebra and its applications
Least squares solution: column of V corresponding to the smallest eigenvalue of A

24. Computing F: The Eight-Point Algorithm
Idea: Compile points into matrix A

25. Computing F: The Eight-Point Algorithm
Decompose A via SVD:
Solution: F is column of V corresponding to the smallest eigenvector of A
In practice: F will be of rank 3, not 2. Correct by
SVD decomposition of F
Set smallest eigenvalue to 0
Reconstruct F’

26. Computing F: The Eight-Point Algorithm
Input: n point correspondences ( n >= 8)
Construct homogeneous system Ax= 0 from
x = (f11,f12, ,f13, f21,f22,f23 f31,f32, f33) : entries in F
Each correspondence give one equation
A is a nx9 matrix
Obtain estimate F^ by SVD of A:
x (up to a scale) is column of V corresponding to the least singular value
Enforce singularity constraint: since Rank (F) = 2
Compute SVD of F:
Set the smallest singular value to 0: D -> D’
Correct estimate of F :
Output: the estimate of the fundamental matrix F’
Similarly we can compute E given intrinsic parameters

27. Recitification Idea: Align Epipolar Lines with Scan Lines.
Question: What type transformation?

28. Locating the Epipoles Input: Fundamental Matrix F
el lies on all the epipolar lines of the left image P Pl Pr p p l r Ol el er Or
Input: Fundamental Matrix F
Find the SVD of F
The epipole el is the column of V corresponding to the null singular value (as shown above)
The epipole er is the column of U corresponding to the null singular value (similar treatment as for el)
Output: Epipole el and er

29. Stereo Rectification (see Trucco)P Pl Pr Yr p p Yl l r Xl Zl Zr T Ol Or Xr
Stereo System with Parallel Optical Axes
Epipoles are at infinity
Horizontal epipolar lines

30. Reconstruction (3-D): IdealizedPl Pr P p p l r Ol Or

31. Reconstruction (3-D): RealPl Pr P p p l r Ol Or
See Trucco/Verri, pages

32. Summary Stereo Vision (Class 1)
Epipolar Geometry: Corresponding points lie on epipolar line
Essential/Fundamental matrix: Defines this line
Eight-Point Algorithm: Recovers Fundamental matrix
Rectification: Epipolar lines parallel to scanlines
Reconstruction: Minimize quadratic distance