A Simplified Algorithm for Solution Classification of the Perspective-three-Point Problem

Given the distance between every pair of 3 control points, and given the angle to every pair of the control points from an additional point called the Center of Perspectivity (P ), find the lengths of the segments joining the P to each of the control points. We call this the “perspective-three-point (p3p) problem”. The corresponding algebraic equation system is called the p3p equation system. So-called a solution classification of the p3p equation system is to give explicit conditions under which the system has none, one, two, ... real physical solutions, respectively. This problem had been open for many years [?, 1-8]. An efficient algorithm is presented here for such a classification under some non-degenerate conditions that gives: • The condition under which the p3p equation system has a unique real physical solution. • The conditions under which the p3p equation system has 2, 3, ... real physical solutions, respectively, provided the system has any real physical solution. The above result gives the complete classificaton (under some non-degenerate conditions) in practice because at least one real solution exists in that case. A program written in Maple was implemented on PC computers efficiently. The problem is originated from camera calibration [?]. Given the distance between every pair of 3 control points, and given the angle to every pair of the control points from an additional point called the Center of Perspectivity (P ), find the lengths of the segments joining the P to each of the control points. We call this the “perspective-three-point (p3p) problem”. The corresponding algebraic equation system  y2 + z2 − 2 p y z − a2 = 0 z2 + x2 − 2 q z x− b2 = 0 x2 + y2 − 2 r x y − c2 = 0 (1) is called the p3p equation system, where a, b, c denote the distances between the control points A,B,C, parameters p, q, r denote the cosines of the angles α = 6 BPC, β = 6 CPA, γ = 6 APB, 1) This work is supported in part by CAS and CNRS under a cooperative project between CICA and LEIBNIZ, and in part by the National Climbing Project and “863” Project of China.