1. The Perspective View of 3 Points
bill wolfe
CSUCI

2. Fischler and Bolles 1981 “Random Sample Consensus” Communications of the ACM, Vol. 24, Number 6, June, 1981
Cartography application
Interpretation of sensed data
Pre-defined models and Landmarks
Noisy measurements
Averaging/smoothing does not always work
Inaccurate feature extraction
Gross errors vs. Random Noise

3. Example (Fischler and Bolles)
RANDSAC
Poison Point
Least Squares

4. Location Determination Problem
image
position and orientation of camera in world frame
camera
world
landmarks
Location Determination Problem

5. Location Determination Problem
Variations:
Pose estimation
Inverse perspective
Camera Calibration
Location Determination
Triangulation

6. Applications Computer Vision Robotics Cartography Computer Graphics
Photogrammetry

7. Assumptions Intrinsic camera parameters are known.
Location of landmarks in world frame are known.
Correspondences between landmarks and their images are known.
Single camera view.
Passive sensing.

8. Strategy
camera
measured/calculated
world
landmarks
known

9. How Many Points are Enough?
1 Point: infinitely many solutions.
2 Points: infinitely many solutions, but bounded.
3 Points:
(no 3 colinear) finitely many solutions (up to 4).
4 Points:
non coplanar (no 3 colinear): finitely many.
coplanar (no 3 colinear): unique solution!
5 Points: can be ambiguous.
6 Points: unique solutions (“general view”).

10. 1 Point

11. 2 Points A CP B

12. Inscribed Angles are Equal http://www. ies. coCP CP q CP q q A B

13. 3 Points s1 s2 s3 LA A B C LB LC

15. Bezout’s Theorem
Number of solutions limited by the product of the degrees of the equations: 2x2x2 = 8.
But, since each term in the equations is of degree 2, each solution L1, L2, L3 generates another solution by taking the negative values -L1, -L2, -L3.
Therefore, there can be at most 4 physically realizable solutions.

16. Algebraic Approach reduce to 4th order equation (Fischler and Bolles, 1981)

17. Iterative Approach s3 s2 s1
slide

18. Iterative Projections

19. Geometric Approach CP

20. The Orthocenter of a Triangle

21. 4 solutions when CP is directly over the orthocenter

22. The Danger Cylinder CP
Why is the Danger Cylinder Dangerous in the P3P Problem? C. Zhang, Z. Hu, Acta Automatiica Sinica, Vol. 32, No. 4, July, 2006.

23. “A General Sufficient Condition of Four Positive Solutions of the P3p Problem”
C. Zhang, Z. Hu, 2005

24. “Complete Solution Classification for the Perspective-Three-Point Problem”
X. Gao, X. Hou, J. Tang, H. Cheng
IEEE Trans PAMI Vol. 25, NO. 8, August 2003

25. 4 Coplanar Points (no 3 colinear)
“Passive Ranging to Known Planar Point Sets”
Y. Hung, P. Yeh, D. Harwood
IEEE Int’l Conf Robotics and Automation, 1985.
Camera CP Q0 Q1 Q3 Q2
Object P0 P1 P3 P2 W L

26. P0_Cam = k0*Q0_Cam P0_Obj = <0,0,0> P1_Cam = k1*Q1_Cam
P1_Obj = <L,0,0>
P2_Obj = <0,W,0>
P3_Obj = <L,W,0>
Camera CP Q0 Q1 Q3 Q2
Object P0 P1 P3 P2
k0*Q0_Cam = P0_Cam W L

27. (P1_Cam - P0_Cam) + (P2_Cam - P0_Cam) = (P3_Cam - P0_Cam)
Camera
Object
P2_Cam
P3_Cam CP
P0_Cam
P1_Cam

28. (P1_Cam - P0_Cam) + (P2_Cam - P0_Cam) = (P3_Cam - P0_Cam)
P0_Cam = k0*Q0_Cam
P1_Cam = k1*Q1_Cam
P2_Cam = k2*Q2_Cam
P3_Cam = k3*Q3_Cam
(P1_Cam - P0_Cam) + (P2_Cam - P0_Cam) = (P3_Cam - P0_Cam)
(k1*Q1_Cam - k0*Q0_Cam) + (k2*Q2_Cam - k0*Q_Cam) =
(k3*Q3_Cam - k0*Q0_Cam)
(k1*Q1_Cam) + (k2*Q2_Cam - k0*Q_Cam) = (k3*Q3_Cam)
let ki’ = ki/k3
(k1’*Q1_Cam) + (k2’*Q2_Cam - k0’*Q0_Cam) = Q3_Cam
k0’*Q0_Cam + k1’*Q1_Cam + k2’*Q2_Cam = Q3_Cam
Three linear equations in the 3 unknowns: k0’, k1’, k2’

29. Camera
Object P0 P1 P3 P2 CP W L
| P3_Obj - P0_Obj | = |P3_Cam - P0_Cam| = | k3*Q3_Cam - k0*Q0_Cam |
k3 = | P3_Obj| / | k0’ * Q0_Cam - Q3_Cam |

30. Orientation Unit_x = (P1_Cam - P0_Cam)/ | P1_Cam - P0_Cam |
Unit_y = (P2_Cam - P0_Cam) / |P2_Cam - P0_Cam|
Unit_z = Unit_x X Unit_y

31. Homgeneous Transformation
Unit_x Unit_y Unit_x P0_Cam 1
H =

32. Summary
Reviewed location determination problems with 1, 2, 3, 4 points.
Algebraic vs. Geometric vs. Iterative methods.
3 points can have up to 4 solutions.
Iterative solution method for 3 points.
4 coplanar points has unique solution.
Complete solution for 4 rectangular points.
Many unsolved geometric issues.

33. References
Random Sample Consensus, Martin Fischler and Robert Bolles, Communications of the ACM, Vol. 24, Number 6, June, 1981
Passive Ranging to Known Planar Point Sets, Y. Hung, P. Yeh, D. Harwood, IEEE Int’l Conf Robotics and Automation, 1985.
The Perspective View of 3 Points, W. Wolfe, D. Mathis, C. Sklair, M. Magee. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 13, No. 1, January 1991.
Review and Analysis of Solutions of the Three Point Perspective Pose Estimation Problem. R. Haralick, C. Lee, K. Ottenberg, M. Nolle. Int’l Journal of Computer Vision, 13, 3, , 1994.
Complete Solution Classification for the Perspective-Three-Point Problem, X. Gao, X. Hou, J. Tang, H. Cheng, IEEE Trans PAMI Vol. 25, NO. 8, August 2003.
A General Sufficient Condition of Four Positive Solutions of the P3P Problem, C. Zhang, Z. Hu, J. Comput. Sci. & Technol., Vol. 20, N0. 6, pp , 2005.
Why is the Danger Cylinder Dangerous in the P3P Problem? C. Zhang, Z. Hu, Acta Automatiica Sinica, Vol. 32, No. 4, July, 2006.