P3P Problem Research Articles

A Novel Solution to the P4P Problem for an Uncalibrated Camera

We present a novel solution for the absolute camera pose and the camera calibration (effective focal length and aspect ratio) based on perspective four point (P4P) problem. By converting perspective transformation to affine transformation and using invariance to 3D affine transformation, we explore the relationship between the dual image of the absolute conic (DIAC) and the world coordinate of camera optical center and show how the coplanar and noncoplanar cases are cast into the problems of solving a quadratic polynomial equation and an eighth degree polynomial equation in a single variable respectively using only linear algebra. In particular, geometric configurations for infinite solutions of the coplanar case are explored. We also confirm the conclusion that the upper bound of eight real solutions for noncoplanar case is attainable by an example. The performance and usefulness of our novel solution are demonstrated by thorough testing on both synthetic and real data.

An efficient algorithm for computing a comprehensive Gröbner system of a parametric polynomial system

A new efficient algorithm for computing a comprehensive Gröbnersystem of a parametric polynomial ideal over k[U][X] is presented. This algorithm generates fewer branches (segments) compared to previously proposed algorithms including Suzuki and Sato’s algorithm as well as Nabeshima’s algorithm. As a result, the algorithm is able to compute comprehensive Gröbnersystems of parametric polynomial ideals arising from applications which have been beyond the reach of other well known algorithms. The starting point of the new algorithm is Weispfenning’s algorithm with a key insight by Suzuki and Sato who proposed computing first a Gröbnerbasis of an ideal over k[U,X] before performing any branches based on parametric constraints. The proposed algorithm exploits the result that along any branch in a tree corresponding to a comprehensive Gröbnersystem, it is only necessary to consider one polynomial for each nondivisible leading power product in k(U)[X] with the condition that the product of their leading coefficients is not 0; other branches correspond to the cases where this product is 0. In addition, for dealing with a disequality parametric constraint, a probabilistic check is employed for radical membership test of an ideal of parametric constraints. This is in contrast to a general expensive check based on Rabinovitch’s trick using a new variable as in Nabeshima’s algorithm. The proposed algorithm has been implemented in Magma and Singular, and experimented with a number of examples from different applications. Its performance (the number of branches and execution time) has been compared with several other existing algorithms. A number of heuristics and efficient checks have been incorporated into the Magma implementation, especially in the case when the ideal of parametric constraints is 0-dimensional. The algorithm has been successfully used to solve a special case of the famous P3P problem from computer vision.

Closed-Form Solution to the Noncoplanar P4P Problem for Uncalibrated Camera

This paper presents a closed-form solution to determination of the position and orientation of a perspective camera with two unknown effective focal lengths for the noncoplanar perspective four point (P4P) problem. Given four noncoplanar 3D points and their correspondences in image coordinate, we convert perspective transformation to affine transformation, and formulate the problem using invariance to 3D affine transformation and arrive to a closed-form solution. We show how the noncoplanar P4P problem is cast into the problem of solving an eighth degree polynomial equation in one unknown. This result shows the noncoplanar P4P problem with two unknown effective focal lengths has at most 8 solutions. Last, we confirm the conclusion by an example. Although developed as part of landmark-guided navigation, the solution might well be used for landmark-based tracking problem, hand-eye coordination, and for fast determination of interior and exterior camera parameters. Because our method is based on closed-form solution, its speed makes it a potential candidate for solving above problems.

An Analytic Solution for the Noncoplanar P4P Problem with An Uncalibrated Camera

Classical PnP problem is based on an important precondition that the intrinsic parameters of a camera are known.But it is more valuable in real applications to investigate the PnP problem with an uncalibrated camera.This paper researches firstly the number of solutions of the noncoplanar P4P problem with an uncalibrated two-parameter pinhole camera,and proposes an analytic solution.The analytic solution can obtain the position and orientation of a camera with respect to world coordinate frame and intrinsic parameter of the camera simultaneously.According to projection equations of four control points and properties of rotation matrix,the analysis is performed first by writing a suitable fourteen equations in fourteen unknowns.Then,by a specifically-developed elimination scheme,the equation set that was consisted of the fourteen equations reduces to an 8th polynomial equation with only one unknown.Last,it is confirmed that the noncoplanar P4P problem with an uncalibrated two-parameter pinhole camera has at most 8 solutions,generally,and also attainable.A large number of synthetic experiments as well as experiments with real images show that the robustness of this method is very strong,and the results are very accurate for fast determination of intrinsic and extrinsic parameters.The algorithm might well be used for object pose estimation,hand-eye coordination and landmark-guided navigation.

A Note on the Roots Distribution and Stability of the PnP Problem

摘要: PnP (Perspectire-n-Point)问题是一种基于单幅图像的定位方法, 由于不需要建立图像点之间的对应关系, 所以, 在机器人定位等应用中得到了广泛的应用. 多解性和解的稳定性是PnP问题的两个重要问题, 直接关系到具体视觉问题的成败. 那么, PnP问题的多解是如何分布的呢? 多解一定意味着解是不稳定的吗? 这些问题文献中几乎没有任何报道. 本文以P3P问题为研究对象, 对这些问题进行了一些探讨, 研究结果对揭示PnP问题解的分布规律以及解的稳定性问题具有一定的参考价值. 关键词: PnP问题 / 多解分布 / 解的稳定性 / 解的唯一性 / 物体定位

A general recursive linear method and unique solution pattern design for the perspective- n-point problem

In this paper, a new method for solving the perspective- n-point (P nP) problem is developed. With an auxiliary point, the linear method for the special case of four coplanar points is extended to find the coarse solutions for the general P3P problem. A recursive least square algorithm with a forgetting factor is introduced to find all the accurate solutions for the P3P problem. Then the algorithm is extended to the general P nP problem. The solution stability issues are investigated for the P3P, P4P and P5P problem, respectively. Furthermore, a pattern is designed to ensure unique solution for the P nP problem. Experiments are performed to verify the effectiveness of the proposed method.

A Probabilistic Study on the Multiple Solutions of the P3P Problem

Under some special conditions,the P3P problem can have 1,2,3 and 4 solutions,and if the 3 control points and the optical center lie on a circle,the problem is indeterminate.In this paper,by the Monte Carlo approach of up to 1 million samples,it is shown that the probabilities of the P3P problem with one solution,two solutions,three solutions,and four solutions are respectively 0.9993,0.0007,0.0000,0.0000.The result confirms the well-known fact that in the most cases,the P3P has a unique solution.

On the Probability of the Number of Solutions for the P4P Problem

This paper studies the multi-solution phenomenon for the perspective four point (P4P) problem from geometric and algebraic aspects. We give a pure geometric proof that the P4P problem could have up to five solutions. We also give a clear picture on how these five solutions could be realized. We prove that with probability one, the P4P problem has a unique solution which can be represented by a set of rational functions in the parameters. The simulant experiments show that to solve the P4P problem with the rational functions is stable and accurate.

PnP Problem Revisited

Perspective-n-Point camera pose determination, or the PnP problem, has attracted much attention in the literature. This paper gives a systematic investigation on the PnP problem from both geometric and algebraic standpoints, and has the following contributions: Firstly, we rigorously prove that the PnP problem under distance-based definition is equivalent to the PnP problem under orthogonal-transformation-based definition when n > 3, and equivalent to the PnP problem under rotation-transformation-based definition when n = 3. Secondly, we obtain the upper bounds of the number of solutions for the PnP problem under different definitions. In particular, we show that for any three non-collinear control points, we can always find out a location of optical center such that the P3P problem formed by these three control points and the optical center can have 4 solutions, its upper bound. Additionally a geometric way is provided to construct these 4 solutions. Thirdly, we introduce a depth-ratio based approach to represent the solutions of the whole PnP problem. This approach is shown to be advantageous over the traditional elimination techniques. Lastly, degenerated cases for coplanar or collinear control points are also discussed. Surprisingly enough, it is shown that if all the control points are collinear, the PnP problem under distance-based definition has a unique solution, but the PnP problem under transformation-based definition is only determined up to one free parameter.

A General Sufficient Condition of Four Positive Solutions of the P3P Problem

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.

Complete solution classification for the perspective-three-point problem

We use two approaches to solve the perspective-three-point (P3P) problem: the algebraic approach and the geometric approach. In the algebraic approach, we use Wu-Ritt's zero decomposition algorithm to give a complete triangular decomposition for the P3P equation system. This decomposition provides the first complete analytical solution to the P3P problem. We also give a complete solution classification for the P3P equation system, i.e., we give explicit criteria for the P3P problem to have one, two, three, and four solutions. Combining the analytical solutions with the criteria, we provide an algorithm, CASSC, which may be used to find complete and robust numerical solutions to the P3P problem. In the geometric approach, we give some pure geometric criteria for the number of real physical solutions.

A note on the number of solutions of the noncoplanar P4P problem

In the literature, the PnP problem is indistinguishably defined as either to determine the distances of the control points from the camera's optical center or to determine the transformation matrices from the object-centered frame to the camera-centered frame. We show that these two definitions are generally not equivalent. In particular, we prove that, if the four control points are not coplanar, the upper bound of the P4P problem under the distance-based definition is 5 and also attainable, whereas the upper bound of the P4P problem under the transformation-based definition is only 4. Finally, we study the conditions under which at least two, three, four, and five different positive solutions exist in the distance based noncoplanar P4P problem.

New algorithms for the Perspective-Three-Point Problem

In this paper, two approaches are used to solve the Perspective-Three-Point Problem (P3P): the symbolic computation approach and the geometric approach. In the symbolic computation approach, we use Wu-Ritt’s zero decomposition algorithm to give a complete triangular decomposition for the P3P equation system. This decomposition provides the first complete analytical solution to the P3P problem. In the geometric approach, we give some pure geometric criteria for the number of real physical solutions. The complete solution classification for two special cases with three and four parameters is also given.

Closed form solution to the twin P4P or the combined three dimensional resection-intersection problem in terms of Möbius barycentric coordinates

The twin perspective 4 point (twin P4P) problem – also called the combined three dimensional resection-intersection problem – is the problem of finding the position of a scene object from 4 correspondence points and a scene stereopair. While the perspective centers of the left and right scene image are positioned by means of a double three dimensional resection, the position of the scene object imaged on the left and right photograph is determined by a three dimensional intersection based upon given resected perspective centers. Here we present a new algorithm solving the twin P4P problem by means of Mobius barycentric coordinates. In the first algorithmic step we determine the distances between the perspective centers and the unknown intersected point by solving a linear system of equations. Typically, area elements of the left and right image build up the linear equation system. The second algorithmic step allows for the computation of the Mobius barycentric coordinates of the unknown intersected point which are thirdly converted into three dimensional object space coordinates {X,Y,Z} of the intersected point. Typically, this three-step algorithm based upon Mobius barycentric coordinates takes advantage of the primary double resection problem from which only distances from four correspondence points to the left and right perspective centre are needed. No orientation parameters and no coordinates of the left and right perspective center have to be made available.

An analytic solution for the perspective 4-point problem

The perspective n-point (P nP) problems is the problem of dinding the position and orientation of a camera with respect to a scene object from n correspondence points. In this paper we propose an analytic solution for the perspective 4-point problem. The solution is found by replacing the four points with a pencil of three lines and by exploring the geometric constraints available with the perspective camera model. We show how the P4P problem is cast into the problem of solving a biquadratic polynomial in one unknown. Although developed as part of an object recognition from a single view system [6], the solution migth well be used for hand-eye coordination, landmark-guided navigation, and for fast determination of exterior camera parameters in general.