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.
The Perspective-3-Point Problem, or P3P problem, is a single-view based pose estimation method. It was first introduced by Grunert [
The P3P problem is defined as follows: Given the perspective projections of three control points with known coordinates in the world system and a calibrated camera, find the position and orientation of the camera in the world system. 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 [
Since, in many real applications, some basic questions must be answered before its real implementation (such as the following: Does it have a unique solution? If not, how many solutions could it have? Is the solution stable?), the multiple solution phenomenon in the P3P problem has been a focus of investigation since its very inception in the literature. Traditionally the multisolution phenomenon in P3P problem is analyzed by at first transforming its 3 quadratic constraints into a quartic equation and then roots of this quartic equation are located to derive possible solutions. For example, Haralick et al. summarized 6 different transformation methods [
In this work, we investigate the multisolution phenomenon in P3P problem for those cases where the camera optical center is very close to one of the 3 toroids. More specifically, based on Grunert’s derivation [
As shown in Figure
P3P problem definition:
As shown in Figure
(a) A single toroid; (b) three toroids.
As shown in [
We have the following main result.
As shown in Figures
If the optical center
Suppose the optical center
Since
When
When
If
In conclusion, if
Since
When the optical center
When
In fact, “the closeness of neighborhood” in the above discussions can be explicitly quantified as
From the above discussion, we can see that if the optical center is close to one of the 3 toroids (inside or outside of the toroid), except for some special curves, there is always a P3P positive solution whose optical center is in a close neighborhood of one of the three control points. This indicates that the 3 control points are singular points of the optical centers distribution of the P3P solutions.
Although the correctness of our proposition lies in its proof, we nevertheless also verify it by computer simulations. Our simulation procedure is as follows.
Generate 3 noncollinear control points
Repeat (1)–(4). Choose at random an optical center From Compute the roots of ( Compute the P3P solution corresponding to
We generate about 10,000 triplets of control points, and for each triplet, about 10,000 different optical centers are simulated. All the simulations confirm our general theoretical conclusion; that is, the optical centers of all the additional P3P solutions are in a close neighborhood of control points. Figure
An example of the resulting optical center distributions around the control point
In this work, we find that, given 3 control points, if the camera optical center is close to any one of the 3 toroids, except for some special curves, there is always an additional positive solution for this P3P problem whose corresponding optical center always lies in a close neighborhood of one of the 3 control points. Hence, the 3 control points are singular points of the optical centers distribution of the P3P solutions. In other words, some singularity exists around the 3 control points for the P3P problem solutions.
Our result provides some new insights into the nature of multisolution phenomenon; in addition to its academic value, the result could also provide some theoretical guidance for P3P practitioners to properly arrange their control points to avoid unstable solutions for the P3P based applications.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by National Natural Science Foundation of China (NSFC) under Grant nos. 61402316 and 61403373.