P3P Problem Research Articles

Relative Localization within a Quadcopter Unmanned Aerial Vehicle Swarm Based on Airborne Monocular Vision

Swarming is one of the important trends in the development of small multi-rotor UAVs. The stable operation of UAV swarms and air-to-ground cooperative operations depend on precise relative position information within the swarm. Existing relative localization solutions mainly rely on passively received external information or expensive and complex sensors, which are not applicable to the application scenarios of small-rotor UAV swarms. Therefore, we develop a relative localization solution based on airborne monocular sensing data to directly realize real-time relative localization among UAVs. First, we apply the lightweight YOLOv8-pose target detection algorithm to realize the real-time detection of quadcopter UAVs and their rotor motors. Then, to improve the computational efficiency, we make full use of the geometric properties of UAVs to derive a more adaptable algorithm for solving the P3P problem. In order to solve the multi-solution problem when less than four motors are detected, we analytically propose a positive solution determination scheme based on reasonable attitude information. We also introduce the maximum weight of the motor-detection confidence into the calculation of relative localization position to further improve the accuracy. Finally, we conducted simulations and practical experiments on an experimental UAV. The experimental results verify the feasibility of the proposed scheme, in which the performance of the core algorithm is significantly improved over the classical algorithm. Our research provides viable solutions to free UAV swarms from external information dependence, apply them to complex environments, improve autonomous collaboration, and reduce costs.

A New Monocular Pose Estimation Method for the Coplanar P4P Problem

The Perspective of N Points (PNP) problem in optical measurement and machine vision is to estimate the pose parameters of an object by a monocular image. For traditional coplanar P4P methods, the four points must have known coordinates or be a rectangle’s four vertices. In this paper, a new method of the P4P problem for parallelogram objects without shape parameters is proposed to solve the objects’ pose and shape parameters based on the parallelogram vertices’ relative coordinate constraints. The correctness and effectiveness of the method have been verified through experiments. In addition to being used to estimate a parallelogram object’s pose and shape, the new method extends the conclusions of the coplanar P4P problem when the four points are a parallelogram’s four vertices.

The Role of the Deltoidal Surface in the Solution Variation of the P3P Problem

The Role of the Deltoidal Surface in the Solution Variation of the P3P Problem

An Algebraic Proof of the Necessary and Sufficient Condition for a P3P Problem Having a Pair of Point-Sharing Solutions

An Algebraic Proof of the Necessary and Sufficient Condition for a P3P Problem Having a Pair of Point-Sharing Solutions

A Novel Method for Intrinsic and Extrinsic Parameters Estimation by Solving Perspective-Three-Point Problem with Known Camera Position

The aim of the perspective-three-point (P3P) problem is to estimate extrinsic parameters of a camera from three 2D–3D point correspondences, including the orientation and position information. All the P3P solvers have a multi-solution phenomenon that is up to four solutions and needs a fully calibrated camera. In contrast, in this paper we propose a novel method for intrinsic and extrinsic parameter estimation based on three 2D–3D point correspondences with known camera position. Our core contribution is to build a new, virtual camera system whose frame and image plane are defined by the original 3D points, to build a new, intermediate world frame by the original image plane and the original 2D image points, and convert our problem to a P3P problem. Then, the intrinsic and extrinsic parameter estimation is to solve frame transformation and the P3P problem. Lastly, we solve the multi-solution problem by image resolution. Experimental results show its accuracy, numerical stability and uniqueness of the solution for intrinsic and extrinsic parameter estimation in synthetic data and real images.

Solving the perspective-three-point problem with linear combinations: An accurate and efficient method

In this paper, we present an efficient algebraic solution to the perspective-three-point (P3P) problem that estimates the pose of a fully-calibrated camera from three given 3D-2D point correspondences. Previous P3P approaches typically solve for two sets of parameters: the depths of three 3D points and the pose of the camera. In contrast, our formulation does not involve any depth factor, and therefore can provide a more compact derivation for the P3P problem. By representing the involved rotation matrix and translation vector as a linear combination of known vectors with unknown coefficients and employing the orthogonal nature of the rotation matrix, this formulation finally leads to a univariate quartic equation which can be solved in closed form. The resulting algorithm is easy to understand and implement as it relies substantially on linear algebra. Experimental results demonstrate that the presented method can achieve a comparable accuracy to the state-of-the-art methods, but with significantly reduced computational requirements.

New insights on multi-solution distribution of the P3P problem

Traditionally, the P3P problem is solved by firstly transforming its 3 quadratic equations into a quartic one, then by locating the roots of the resulting quartic equation and verifying whether a root does really correspond to a true solution of the P3P problem itself. It is well known that a root of the quartic equation could correspond to 2, or 1, or even null solution at all to the P3P problem, and up to now, no explicit relationship between the P3P solution and the root of its quartic equation is available in the literature. In this work, we show that when the optical center is outside of all the 6 toroids defined by the control point triangle, each positive root of the Grunert's quartic equation must correspond to a true solution of the P3P problem, and the corresponding P3P problem cannot have a unique solution, it must have either 2 positive solutions or 4 positive solutions. In addition, we show that when the optical center passes through any one of the 3 toroids among these 6 toroids (except possibly for two concentric circles), the number of the solutions of the corresponding P3P problem always changes by 1, either increased by 1 or decreased by 1. Furthermore we show that such changed solutions always locate in a small neighborhood of control points, hence the 3 toroids are critical surfaces of the P3P problem and the 3 control points are 3 singular points of solutions. A notable example is that when the optical center passes through the outer surface of the union of the 6 toroids from the outside to inside, the number of the solutions must always decrease by 1. Our results are the first in the literature to give an explicit and geometrically intuitive relationship between the P3P solutions and the roots of its quartic equation, aside its academic values, it could also act as some theoretical guidance for P3P practitioners to properly arrange their control points to avoid undesirable solutions.

Geometric Interpretation of the Multi-solution Phenomenon in the P3P Problem

It is well known that the P3P problem could have 1, 2, 3 and at most 4 positive solutions under different configurations among its three control points and the position of the optical center. Since in any real applications, the knowledge on the exact number of possible solutions is a prerequisite for selecting the right one among all the possible solutions, and the study on the phenomenon of multiple solutions in the P3P problem has been an active topic since its very inception. In this work, we provide some new geometric interpretations on the multi-solution phenomenon in the P3P problem, and our main results include: (1) the necessary and sufficient condition for the P3P problem to have a pair of side-sharing solutions is the two optical centers of the solutions both lie on one of the three vertical planes to the base plane of control points; (2) the necessary and sufficient condition for the P3P problem to have a pair of point-sharing solutions is the two optical centers of the solutions both lie on one of the three so-called skewed danger cylinders;(3) if the P3P problem has other solutions in addition to a pair of side-sharing (point-sharing) solutions, these remaining solutions must be a point-sharing (side-sharing ) pair. In a sense, the side-sharing pair and the point-sharing pair are companion pairs; (4) there indeed exist such P3P problems that have four completely distinct solutions, i.e., the solutions sharing neither a side nor a point, closing a long guessing issue in the literature. In sum, our results provide some new insights into the nature of the multi-solution phenomenon in the P3P problem, and in addition to their academic value, they could also be used as some theoretical guidance for practitioners in real applications to avoid occurrence of multiple solutions by properly arranging the control points.

A single-shot pose estimation approach for a 2D laser rangefinder

Two-dimensional (2D) laser rangefinders (LRFs) are widely utilized in various mobilized systems, such as robots, intelligent vehicles (IVs) and mobile mapping. Pose estimation for a 2D LRF has important applications for exterior calibration of multiple sensors, registering light detection and ranging (LIDAR) data from multiple LRFs or from a spinning/nodding LRF, etc. This paper shows that the pose of a 2D LRF can be uniquely determined from a single-shot of a trirectangular trihedron. The pose is estimated by solving a simplified perspective-three-point (P3P) problem and a three-point-registration problem. Compared to the raw noisy LIDAR data, the three control points in the P3P problem are estimated from line fitting of the sensed LIDAR data to guarantee good and reliable estimation results. The proposed method has been validated with both simulation and real data. The results show that the method is accurate and practical. To the best of our knowledge, this paper solves the pose estimation of a 2D LRF by using LIDAR data from one scan for the first time.

A LiDAR data-based camera self-calibration method

To find the intrinsic parameters of a camera, a LiDAR data-based camera self-calibration method is presented here. Parameters have been estimated using particle swarm optimization (PSO), enhancing the optimal solution of a multivariate cost function. The main procedure of camera intrinsic parameter estimation has three parts, which include extraction and fine matching of interest points in the images, establishment of cost function, based on Kruppa equations and optimization of PSO using LiDAR data as the initialization input. To improve the precision of matching pairs, a new method of maximal information coefficient (MIC) and maximum asymmetry score (MAS) was used to remove false matching pairs based on the RANSAC algorithm. Highly precise matching pairs were used to calculate the fundamental matrix so that the new cost function (deduced from Kruppa equations in terms of the fundamental matrix) was more accurate. The cost function involving four intrinsic parameters was minimized by PSO for the optimal solution. To overcome the issue of optimization pushed to a local optimum, LiDAR data was used to determine the scope of initialization, based on the solution to the P4P problem for camera focal length. To verify the accuracy and robustness of the proposed method, simulations and experiments were implemented and compared with two typical methods. Simulation results indicated that the intrinsic parameters estimated by the proposed method had absolute errors less than 1.0 pixel and relative errors smaller than 0.01%. Based on ground truth obtained from a meter ruler, the distance inversion accuracy in the experiments was smaller than 1.0 cm. Experimental and simulated results demonstrated that the proposed method was highly accurate and robust.

A P3P problem solver representing all parameters as a linear combination

We propose a novel strategy for the Perspective-Three-Point (P3P) problem that determines the position and orientation of a calibrated camera from three known point pairs of 2D-3D correspondences. Starting from three similarity transformation equations that relate the global and the camera-oriented coordinates, our method treats all the extrinsic camera parameters as a linear combination of known vectors with unknown coefficients. By reducing the number of unknowns and using a Gröbner basis, the problem is converted into a fourth-order polynomial equation with a single unknown parameter. Experimental results show that our method is highly practical and precise. Moreover, the performance of our method and its robustness to image noise are similar to those of a state-of-the-art method. In addition, our method exhibits greater computational efficiency than other methods.

An efficient solution to the perspective-three-point pose problem

In this paper, we present a new algebraic method to solve the perspective-three-point (P3P) problem, which directly computes the rotation and position of a camera in the world frame without the intermediate derivation of the points in the camera frame. Unlike other online methods, the proposed method uses an “object” coordinate frame, in which the known 3D coordinates are sparse, facilitating formulations of the P3P problem. Additionally, two auxiliary variables are introduced to parameterize the rotation and position matrix, and then a closed-form solution for the camera pose is obtained from subsequent substitutions. This algebraic approach makes the processes more easily followed and significantly improves the performance. Experimental results demonstrated that our method offers accuracy and precision comparable to the existing state-of-the art methods but it has better computational efficiency.

Generality of the multi-solution phenomenon in the P3P problem

The P3P problem is a significant single-view-based pose estimation method, and its multi-solution phenomenon has been the focus of investigation since its inception. In this work, we demonstrate from a geometrical point of view that when the distance from the cameras optical center to the three control points is relatively large, the corresponding P3P problem cannot have a unique solution in general; alternatively, or it should have at least two solutions satisfying the three original constraints. A further contribution of this study is the finding that the intersection of the three 3D pumpkin surfaces in the P3P problem can be generally deduced from the intersection of the three 2D intersecting curves of the three pumpkin surfaces with the control point plane, which provides a new way to investigate the multi-solution phenomenon. Our results may be significant for both P3P theorists and practitioners.

Pose Estimation from Line Correspondences: A Complete Analysis and a Series of Solutions.

In this paper we deal with the camera pose estimation problem from a set of 2D/3D line correspondences, which is also known as PnL (Perspective-n-Line) problem. We carry out our study by comparing PnL with the well-studied PnP (Perspective-n-Point) problem, and our contributions are three-fold: (1) We provide a complete 3D configuration analysis for P3L, which includes the well-known P3P problem as well as several existing analyses as special cases. (2) By exploring the similarity between PnL and PnP, we propose a new subset-based PnL approach as well as a series of linear-formulation-based PnL approaches inspired by their PnP counterparts. (3) The proposed linear-formulation-based methods can be easily extended to deal with the line and point features simultaneously.

Extrinsic Calibration of 2-D Laser Rangefinder and Camera From Single Shot Based on Minimal Solution

A 2-D laser rangefinder (LRF) and a camera are widely applied in various mobilized systems, such as autonomous robot and unmanned ground vehicle. Integration of these two types of sensors promotes a new measurement technology. Extrinsic calibration is crucial and basically the first step to fuse image and laser data. The chessboard-plane-based method is one of the most popular ones, which requires at least five plane-input (five shots). Vasconcelos et al. recently reduced the number of shots from five to three and derived the minimal solution. However, it requires solving a sophisticated perspective-three-point (P3P) problem with eight solutions. This paper first introduces a virtual trihedron from three plane-input and reformulates the P3P problem directly in 3-D space rather than in 3-D dual space to derive the minimal solution for extrinsic calibration. Based on this, we propose a novel and flexible method for extrinsic calibration of an LRF–camera with a trirectangular trihedron (or a cube). One unique feature is that the method requires only a single shot of the target, which can greatly simplify the calibration procedure. The proposed calibration method involves solving simplified P3P and perspective-three-line (P3L) problems separately. In particular, computation of LRF pose is formulated as a simplified P3P problem, which is easily solved and has unique solution from single-shot laser data only. It also avoids the degeneration problem. Camera pose computation is formulated as a simplified P3L problem from image data. The proposed calibration method has been tested with both simulation and real data. The results show that the method is both accurate and flexible for extrinsic calibration of a 2-D LRF and a camera.

Solving the perspective-three-point problem using comprehensive Gröbner systems

A complete solution classification of the perspective-three-point (P3P) problem is given by using the Grobner basis method. The structure of the solution space of the polynomial system deduced by the P3P problem can be obtained by computing a comprehensive Grobner system. Combining with properties of the generalized discriminant sequences, the authors give the explicit conditions to determine the number of distinct real positive solutions of the P3P problem. Several examples are provided to illustrate the effectiveness of the proposed conditions.

Singular Points in the Optical Center Distribution of P3P Solutions

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.

Solution Classification for Perspective-Three-Point Problem Base on PST Method

The perspective-n-point (PnP) problem is originated from camera calibration. It is to determine the position and orientation of the camera with respect to a scene object from n correspondent points. And a new stable algorithm by using a geometric constraint called perspective similar triangle (PST) can give new equations to solve P3P. The PST method achieves high stability in the permutation problem and in presence of image noise. Using the complete discrimination system, we obtain the solution classification of the new equation for the P3P problem. The solution classification gives a set of formulas to determine the number of real solutions to the P3P problem. Based on the formulas, we may know whether the parameters give multiple solutions or not and are critical or not which is very important to present robust algorithm.

Calibrating distance mapping of non‐SVP catadioptric camera of the soccer robot

PurposeThe purpose of the present paper is to propose a full model‐based method for distance‐mapping calibration for the non‐SVP (non‐single viewpoint) catadioptric camera of the soccer robot. The method should be easy to operate, efficient, accurate, and scalable to fit larger field sizes.Design/methodology/approachThe distance‐mapping model was first constructed based on the imaging principle. The authors then calibrated the internal parameters using the mirror boundary and used the mirror center to choose the correct pose from two possible solutions. The authors then proposed a three‐point method based on a unique solution case of the non‐SVP P3P (perspective‐three‐point) problem to solve the external parameters. Lastly, they built the distance mapping by back‐projection.FindingsThe simulation experimental results have shown that the authors' method is very accurate even when there is severe misalignment between the mirror and the camera and that all calibration operations, except the calibration of a standard camera, can be completed in 1 min. The result of the comparison with the traditional calibration method shows that the authors' method is superior to the traditional method in terms of accuracy and efficiency.Originality/valueThe proposed calibration method is scalable to larger fields because it only uses the boundary of the mirror and three feature points on the field, and does not need additional calibration objects. Additionally, an automatic calibration method that can be used during the game can be easily developed based on this method. Moreover, the proposed mirror‐pose‐selection method and a unique solution to the non‐SVP P3P problem are especially useful for a non‐SVP catadioptric camera.

A Minimal Solution for the Extrinsic Calibration of a Camera and a Laser-Rangefinder

This paper presents a new algorithm for the extrinsic calibration of a perspective camera and an invisible 2D laser-rangefinder (LRF). The calibration is achieved by freely moving a checkerboard pattern in order to obtain plane poses in camera coordinates and depth readings in the LRF reference frame. The problem of estimating the rigid displacement between the two sensors is formulated as one of registering a set of planes and lines in the 3D space. It is proven for the first time that the alignment of three plane-line correspondences has at most eight solutions that can be determined by solving a standard p3p problem and a linear system of equations. This leads to a minimal closed-form solution for the extrinsic calibration that can be used as hypothesis generator in a RANSAC paradigm. Our calibration approach is validated through simulation and real experiments that show the superiority with respect to the current state-of-the-art method requiring a minimum of five input planes.