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A Note on the Number of Solutions of the Noncoplanar P4P Problem
April 2002 (vol. 24 no. 4)
pp. 550-555

In the literature, the PnP problem is indistinguishably defined as either to determine the distances of the control points from the camera's optical center or to determine the transformation matrices from the object-centered frame to the camera-centered frame. In this paper, we show that these two definitions are generally not equivalent. In particular, we prove that, if the four control points are not coplanar, the upper bound of the P4P problem under the distance-based definition is 5 and also attainable, whereas the upper bound of the P4P problem under the transformation-based definition is only 4. Finally, we study the conditions under which at least two, three, four, and five different positive solutions exist in the distance based noncoplanar P4P problem.

[1] M.A. Fischler and R.C. Bolles, “Random Sample Consensus: A Paradigm for Model Fitting with Applications to Image Analysis and Automated Cartography,” Graphics and Image Processing, vol. 24, no. 6, pp. 381–395, June 1981.
[2] J. Huang, J.A. Stankovic, K. Ramamritham, D. Towsley, and B. Purimetla, “On Using Priority Inheritance in Real-Time Databases,” Special Issue of Real-Time Systems J., vol. 4. no. 3, Sept. 1992.
[3] R. Horaud, B. Conio, O. Leboulleux, and B. Lacolle, “An Analytic Solution for the Perspective 4-Point Problem,” Computer Vision, Graphics, and Image Processing, vol. 47, pp. 33–44, 1989.
[4] W.J. Wolfe and D. Mathis, “The Perspective View of Three Points,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 1, pp. 66-73, 1991.
[5] C. Su, Y.Q. Xu, H. Li, and S.Q. Liu, “Necessary and Sufficient Condition of Positive Root Number of Perspective-Tree-Point Problem,” Chinese J. Computers, (in Chinese), vol. 21, no. 12, pp. 1084-1095, 1998.
[6] R.M. Haralick, “Determining Camera Parameters from the Perspective Projection of a Rectangle,” Pattern Recognition, vol. 22, no. 3, pp. 225-230, 1989.
[7] M.A. Penna, “Determining Camera Parameters from the Perspective Projection of a Quadrilateral,” Pattern Recognition, vol. 24, no. 6, pp. 533-541, 1991.
[8] L. Quan and Z.D. Lan, “Linear${\rm{N}}>= 4$Point Pose Determination,” Proc. Int'l Conf. Pattern Recognition, pp. 778-783, 1998.
[9] M.L. Liu and K.H. Wong, “Pose Estimation Using Four Corresponding Points,” Pattern Recognition Letters 20, pp. 69-74, 1999.
[10] M. Dhome, M. Richetin, J.T. Lapreste, and G. Rives, “Determination of the Attitude of 3D Objects from a Single Perspective View,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, no. 12, pp. 1,265-1,278, Dec. 1989.
[11] M.A. Abidi and T. Chandra, “A New Efficient and Direct Solution for Pose Estimation Using Quadrangular Targets: Algorithm and Evaluation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, no. 5, pp. 534-538, 1995.
[12] R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision. Cambridge Univ. Press, 2000.
[13] E.H. Thompson, “Space Resection: Failure Cases,” Photogrammetric Record, vol. X, no. 27, pp. 201-204, 1966.
[14] A.D.N. Smith, “The Explicit Solution of Single Picture Resection Problem with a Least Squares Adjustment to Redundant Control,” Photogrammetric Record, vol. V, no. 26, pp. 113-122, 1965.
[15] K. Georhis, M. Petrou, and J. Kittler, “Error Guided Design of a 3D Vision System,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 20, no. 4, pp. 366-379, 1998.

Index Terms:
The Noncoplanar P4P Problem, rigid transformation, upper bound
Citation:
Z.Y. Hu, F.C. Wu, "A Note on the Number of Solutions of the Noncoplanar P4P Problem," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 24, no. 4, pp. 550-555, April 2002, doi:10.1109/34.993561
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