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The Perspective View of 3 Points

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

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The Perspective View of 3 Points

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  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. 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 landmarks world 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 Equalhttp://www.ies.co.jp/math/java/geo/enshukaku/enshukaku.html CP CP q CP q q A B

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

  14. 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.

  15. Algebraic Approachreduce to 4th order equation(Fischler and Bolles, 1981) http://planetmath.org/encyclopedia/QuarticFormula.html

  16. s3 Iterative Approach s2 s1 slide

  17. Iterative Projections http://faculty.csuci.edu/william.wolfe/csuci/articles/TNN_Perspective_View_3_pts.pdf

  18. CP Geometric Approach

  19. The Orthocenter of a Triangle http://www.mathopenref.com/triangleorthocenter.html

  20. 4 solutions when CP is directly over the orthocenter CP

  21. 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.

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

  23. “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

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

  25. Camera Q2 CP P2 Q3 Q0 P3 Q1 P0 P1 P0_Obj = <0,0,0> P1_Obj = <L,0,0> P2_Obj = <0,W,0> P3_Obj = <L,W,0> P0_Cam = k0*Q0_Cam P1_Cam = k1*Q1_Cam P2_Cam = k2*Q2_Cam P3_Cam = k3*Q3_Cam Object k0*Q0_Cam = P0_Cam W L

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

  27. 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’

  28. Camera P2 P3 P0 P1 Object 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 |

  29. 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

  30. Unit_x Unit_y Unit_x P0_Cam 0 0 0 1 Homgeneous Transformation H =

  31. 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.

  32. 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, 331-356, 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. 836-842, 2005. • Why is the Danger Cylinder Dangerous in the P3P Problem? C. Zhang, Z. Hu, Acta Automatiica Sinica, Vol. 32, No. 4, July, 2006.

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