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Journal of Mathematical Imaging and Vision

, Volume 51, Issue 2, pp 326–337 | Cite as

Mathematical Analysis of the Multisolution Phenomenon in the P3P Problem

  • M. VynnyckyEmail author
  • K. Kanev
Article

Abstract

The perspective 3-point problem, also known as pose estimation, has its origins in camera calibration and is of importance in many fields: for example, computer animation, automation, image analysis and robotics. One line of activity involves formulating it mathematically in terms of finding the solution to a quartic equation. However, in general, the equation does not have a unique solution, and in some situations there are no solutions at all. Here, we present a new approach to the solution of the problem; this involves closer scrutiny of the coefficients of the polynomial, in order to understand how many solutions there will be for a given set of problem parameters. We find that, if the control points are equally spaced, there are four positive solutions to the problem at 25 % of all available spatial locations for the control-point combinations, and two positive solutions at the remaining 75 %.

Keywords

P3P Quartic polynomial Multiple solutions 

Notes

Acknowledgments

M. Vynnycky acknowledges the support of the Mathematics Applications Consortium for Science and Industry www.macsi.ul.ie, funded by the Science Foundation Ireland (SFI) Mathematics Initiative Grant 06/MI/005 and SFI Grant 12/IA/1683, as well as funding support from Research Institute of Electronics, Shizuoka University, Japan, for visiting and cooperative research. The authors also acknowledge useful discussions with Prof. J. A. Cuminato, and the comments of an anonymous referee.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Mathematics Applications Consortium for Science and Industry (MACSI), Department of Mathematics and StatisticsUniversity of LimerickLimerickRepublic of Ireland
  2. 2.Division of Casting of Metals, Department of Materials Science and EngineeringRoyal Institute of TechnologyStockholmSweden
  3. 3.Research Institute of ElectronicsShizuoka UniversityHamamatsuJapan

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