The Perspective View of 3 Points bill wolfe CSUCI

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

Example (Fischler and Bolles) Least Squares RANDSAC Poison Point

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

Variations: –Pose estimation –Inverse perspective –Camera Calibration –Location Determination –Triangulation

Applications Computer Vision Robotics Cartography Computer Graphics Photogrammetry

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.

Strategy camera world landmarks measured/ calculated known

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”).

1 Point

2 Points CP A B

Inscribed Angles are Equal    CP A B

3 Points s1 s2 s3 LA A B C LB LC

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.

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

Iterative Approach s1 s2 slide s3

Iterative Projections

Geometric Approach CP

The Orthocenter of a Triangle

CP 4 solutions when CP is directly over the orthocenter

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

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

“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

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, P0 P1 P3 P2 L W Object Camera CP Q0 Q1 Q3 Q2

P0_Obj = P1_Obj = P2_Obj = P3_Obj = P0_Cam = k0*Q0_Cam P1_Cam = k1*Q1_Cam P2_Cam = k2*Q2_Cam P3_Cam = k3*Q3_Cam P0 P1 P3 P2 L W Object Camera CP Q0 Q1 Q3 Q2 k0*Q0_Cam = P0_Cam

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

(k1*Q1_Cam - k0*Q0_Cam) + (k2*Q2_Cam - k0*Q_Cam) = (k3*Q3_Cam - k0*Q0_Cam) (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 (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’

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

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 Orientation

Homgeneous Transformation Unit_x Unit_y Unit_x P0_Cam H =

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.

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