A general sufficient condition of four positive solutions of the P3P problem

Authors:
Cai-Xia Zhang;Zhan-Yi Hu
Affiliations:
National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, Beijing, P.R. China;National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, Beijing, P.R. China
Venue:
Journal of Computer Science and Technology
Year:
2005

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Abstract

In this paper, it is proved that, given 3 control points A, B and C, if the camera's optical center O lies on one of the three planes perpendicular to the plane ABC and going through one of the three altitudes of the triangle ABC, and additionally its projection on the plane ABC is within the circumscribed circle of the triangle, that is, O is within the so-called "danger cylinder", then the corresponding P3P problem {O, (ABC)} must have 4 positive solutions. This result is purely geometrical, and more instructive. It can bring some new insight into a better understanding of multiple-solution problem in the PnP problem, and could be used as some theoretical guide to arrange control points in real applications.