Abstract
Multisolution phenomenon is an important issue in P3P problem since, for many real applications, the question of how many solutions could possibly exist for a given P3P problem must at first be addressed before any real implementation. In this work we show that, given 3 control points, if the camera’s optical center is close to one of the 3 toroids generated by rotating the circumcircle of the control point triangle around each one of its 3 sides, there is always an additional solution with its corresponding optical center lying in a small neighborhood of one of the control points, in addition to the original solution. In other words, there always exist at least two solutions for the P3P problem in such cases. Since, for all such additional solutions, their corresponding optical centers must lie in a small neighborhood of control points, the 3 control points constitute the singular points of the P3P solutions. The above result could act as some theoretical guide for P3P practitioners besides its academic value.
Highlights
IntroductionThe Perspective-3-Point Problem, or P3P problem, is a singleview based pose estimation method
The Perspective-3-Point Problem, or P3P problem, is a singleview based pose estimation method. It was first introduced by Grunert [1] in 1841 and popularized in computer vision community a century later by mainly Fischler and Bolles’ work in 1981 [2]. Since it is the least number of points to have a finite number of solutions and no feature-matching across views is needed, it has been widely used in various fields [3,4,5,6,7,8,9,10,11], either for its minimal demand in restricted working environment, such as robotics and aeronautics, or for its computational efficiency acting as a minimum-set based solver repeatedly called in robust-statistics based iterative estimation framework, such as the well-known RANSAC-like framework, where the computational time increases exponentially with the cardinality of the minimum set; the P3P problem is preferred due to its minimum requirement
It is shown that the P3P problem could have 1, 2, 3, or at most 4 solutions depending on the configuration between the camera optical center and its 3 control points [12]
Summary
The Perspective-3-Point Problem, or P3P problem, is a singleview based pose estimation method. Based on Grunert’s derivation [12], we show that when the optical center of a given P3P problem is close to any one of the 3 toroids generated by rotating the circumcircle of the control point triangle around each of its 3 sides, in addition to its original solution, there is always an additional solution whose optical center always lies in a small neighborhood of control points. In other words, such a P3P problem always has at least two solutions. “two different solutions” refer to “the two solutions with the same 3 control points, but their optical centers lie at a different position.”
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