Abstract
The perspective-n-point (PnP) problem is to find the position and orientation of a camera with respect to a scene object from n correspondence points and is a widely used technique for pose determination in the computer vision community. Finding out geometric conditions of multiple solutions is the ultimate and most desirable goal of the multi-solution analysis, a key research issue of the problem in the literature. In this paper, we study the multi-solution phenomenon of the P4P problem and give some necessary conditions under which there are five positive solutions for the P4P problem. Moreover, we give a geometric configuration for the five solutions.
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Abidi, M.A., Chandra, T.: A New Efficient and Direct Solution for Pose Estimation Using Quadrangular Targets: Algorithm and Evaluation. IEEE Transaction on Pattern Analysis and Machine Intelligence 17(5), 534–538 (1995)
Abdel-Aziz, Y.I., Karara, H.M.: Direct Linear Transformation into Object Space Coordinates in Close-range Photogrammetry. In: Proc. Symp. Close-range Photogrammetry, pp. 1–18 (1971)
Ansar, A., Daniilidis, K.: Linear Pose Estimation from Points and Lines. IEEE Transaction on Pattern Analysis and Machine Intelligence 25(5), 578–589 (2003)
DeMenthon, D., Davis, L.S.: Exact and Approximate Solutions of the Perspective-Three-Point Problem. IEEE Transaction on Pattern Analysis and Machine Intelligence 14(11), 1100–1105 (1992)
DeMenthon, D.F., Davis, L.S.: Model-Based Object Pose in 25 Lines of Code. International Journal of Computer Vision 15, 123–141 (1995)
Fischler, M.A., Bolles, R.C.: Random Sample Consensus: A Paradigm for Model Fitting with Applications to Image Analysis and Automated Cartomated Cartography. Communications of the ACM 24(6), 381–395 (1981)
Ganapathy, S.: Decomposition of Transformtion Matrices for Robot Vision. In: Proc. IEEE Con. Robotics and Automation, pp. 130–139. IEEE Press, Los Alamitos (1984)
Gao, X.S., Hou, X.R., Tang, J.L., Cheng, H.: Complete Solution Classification for the Perspective-Three-Point Problem. IEEE Transaction on Pattern Analysis and Machine Intelligence 25(8), 534–538 (2003)
Gao, X.S., Tang, J.L.: On the Solution Number of Solutions for the P4P Problem. Mathematics-Mechanization Research Center Preprints Preprint 21, 64–76 (2002)
Haralick, R.M., Lee, C., Ottenberg, K., Nolle, M.: Analysis and Solutions of the Three Point Perspective Pose Estimation Problem. In: Proc. of the Int. Conf. on Computer Vision and Pattern Recognition, pp. 592–598 (1991)
Hartley, R.I., Zisserman, A.: Multiple view geometry in computer vision. Cambridge University Press, Cambridge (2000)
Horaud, R., Conio, B., Leboulleux, O.: An Analytic Solution for the Perspective 4-Point Problem. Computer Vision, Graphics and Image Processing 47, 33–44 (1989)
Horn, B.K.P.: Closed Form Solution of Absolute Orientation Using Unit Quaternions. Journal of the Optical Society of America 5(7), 1127–1135 (1987)
Hu, Z.Y., Wu, F.C.: A Note on the Number Solution of the Non-coplanar P4P Problem. IEEE Transactions on Pattern Analysis and Machine Intelligence 24(4), 550–555 (2002)
Hu, Z.Y., Wu, F.C.: A Study on the P5P Problem. Chinese Journal of Software 12(5), 768–775 (2001) (in Chinese)
Mishra, B.: Algorithmic Algebra. Springer, New York (1993)
Pehkonen, K., Harwood, D., Davis, L.S.: Parallel Calculation of 3-D Pose of a Known Object in a Single View. Pattern Recognition Lett. 12, 353–361 (1991)
Quan, L., Lan, Z.: Linear N-Point Camera Pose Determination. IEEE Transaction on Pattern Analysis and Machine Intelligence 21(8), 774–780 (1999)
Reid, G.J., Tang, J., Zhi, L.: A complete symbolic-numeric linear method for camera pose determination. In: Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, Scotland. ACM Press, New York (2003)
Rives, P., Bouthémy, P., Prasada, B., Dubois, E.: Recovering the Orientation and the Position of a Rigid Body in Space from a Single View. Technical Report INRS-Télécommunications place du commerce Ile-des-Soeurs Verdun H3E 1H6 Quebec Canada 3 (1981)
Su, C., Xu, Y., Li, H., Liu, S.: Necessary and Sufficient Condition of Positive Root Number of P3P Problem. Chinese Journal of Computer Sciences 21, 1084–1095 (1998) (in Chinese)
Su, C., Xu, Y., Li, H., Liu, S.: Application of Wu’s Method in Computer Animation. In: The Fifth Int. Conf. on Computer Aided Design/Computer Graphics, vol. 1, pp. 211–215 (1997)
Wang, D., Gao, X.-S.: Counting the Number of Solution for Algebraic Parametric Equation Systems. Mathematics-Mechanization Research Center Priprints 20, 209–220 (2001)
Wolfe, W.J., Jones, K.: Camera Calibration Using the Perspective View of a Triangle. In: Proc. SPIE Conf. Auto. Inspection Measurement, vol. 730, pp. 47–50 (1986)
Wu, W.T.: Basic principles of mechanical theorem proving in geometries: Part of Elementary Geometries, vol. I. Science Press, Beijing (1984) (in Chinese) (English Version Springer-Verlag 1995)
Wu, W.T.: Mathematics Mechanization. Science Press, Beijing (2000) (in Chinese); English Version Kluwer Aacademic Publishers London (2000)
Yuan, J.S.C.: A General Photogrammetric Method for Determining Object Position and Orientation. IEEE Transaction on Robotics and Automation 5(2), 129–142 (1989)
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Tang, J. (2005). Some Necessary Conditions on the Number of Solutions for the P4P Problem. In: Li, H., Olver, P.J., Sommer, G. (eds) Computer Algebra and Geometric Algebra with Applications. IWMM GIAE 2004 2004. Lecture Notes in Computer Science, vol 3519. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11499251_6
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DOI: https://doi.org/10.1007/11499251_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26296-1
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