Mathematical Problems in Engineering

Volume 2015, Article ID 147397, 6 pages

http://dx.doi.org/10.1155/2015/147397

## Singular Points in the Optical Center Distribution of P3P Solutions

^{1}School of Computer Science and Technology, Taiyuan University of Science and Technology, Taiyuan 030024, China^{2}Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China

Received 27 April 2015; Revised 15 June 2015; Accepted 21 June 2015

Academic Editor: Erik Cuevas

Copyright © 2015 Lihua Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

The Perspective-3-Point Problem, or P3P problem, is a single-view 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 [3–11], 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; hence, the P3P problem is preferred due to its minimum requirement.

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 [12]. If the optical center and 3 control points happen to be concyclic, the problem becomes degenerated, and an infinitely large number of solutions could be possible.

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 [12]. Gao et al. [13] gave a complete solution classification. Gao and Tang [14] also gave an analysis on the solutions distribution from the probabilistic point of view. Recently, Rieck [15–17] gave a systematic analysis on the multisolution phenomenon, in particular, on the cases where the optical center of the P3P problem is close to the danger cylinder. From the geometric point of view, Zhang and Hu [18] showed that when the optical center lies on the danger cylinder, the corresponding P3P problem must have 3 distinct solutions, and one must be a double solution. Lowe [3] and Zhang and Hu [19] showed that when the optical center lies on the three vertical planes perpendicular to the control point plane and going through one of the 3 altitudes of the control point triangle, the P3P problem must have a pair of side-sharing solutions and a pair of point-sharing solutions. (As shown in [19], a pair of side-sharing solutions refers to two such solutions that if their optical centers are superposed, they must share two control points or share a common side of the control point triangle. Similarly, a pair of point-sharing solutions refers to two such solutions that if their optical centers are superposed, they must share a single common control point.) Sun and Wang [20] gave an interpretation of the solution changes at the intersecting lines of the above three vertical planes with the danger cylinder. Wu and Hu [21] gave a thorough investigation on the degenerate cases in their “The PnP Problem Revisited” work. Recently, Wang et al. [22] show that the side-sharing pair of solutions is usually accompanied by a point-sharing pair of solutions, or the two kinds of pairs are often companion pairs. They also show [23] that if the optical center is outside of all the 6 toroids, each positive root of Grunert’s quartic equation must correspond to a positive solution of the P3P problem. Their work shows that the root-solution-relationship is of one-to-one correspondence when the optical center is outside of all the 6 toroids but much more complicated when the optical center is inside of them.

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 [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. In addition, since, for all such additional solutions, their optical centers must be in a close neighborhood of control points, the 3 control points are singular points of the solutions’ optical centers distribution.

#### 2. P3P Problem, 3 Toroids, and Grunert’s Quartic Equation

##### 2.1. P3P Problem

As shown in Figure 1, , , and are the three control points with known distance: , , and ; is the camera optical center. Since the camera (under the pinhole model) is calibrated, the three subtended angles , , and of the projection rays can be considered as known entities and then, by the Law of Cosines, the following 3 basic constraints on the three unknowns , , and in (1) must hold: Hence, the P3P problem means to determine such positive triplets satisfying all the 3 basic constraints in (1). And the multisolution phenomenon refers to the existence of multiple such positive triplets all satisfying the 3 constraints in (1).