This paper presents a general solution to the determination of the pose of a perspective camera with unknown focal length from images of four 3D reference points. Our problem is a generalization of the P3P and P4P problems previously developed for fully calibrated cameras. Given four 2D-to-3D correspondences, we estimate camera position, orientation and recover the camera focal length. We formulate the problem and provide a minimal solution from four points by solving a system of algebraic equations. We compare the Hidden variable resultant and Gröbner basis techniques for solving the algebraic equations of our problem. By evaluating them on synthetic and on real-data, we show that the Gröbner basis technique provides stable results.

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Epipolar geometry and relative camera pose computation for uncalibrated cameras with radial distortion has recently been formulated as a minimal problem and successfully solved in floating point arithmetics. The singularity of the fundamental matrix has been used to reduce the minimal number of points to eight. It was assumed that the cameras were not calibrated but had same distortions. In this paper we formulate two new minimal problems for estimating epipolar geometry of cameras with radial distortion. First we present a minimal algorithm for partially calibrated cameras with same radial distortion. Using the trace constraint which holds for the epipolar geometry of calibrated cameras to reduce the number of necessary points from eight to six. We demonstrate that the problem is solvable in exact rational arithmetics. Secondly, we present a minimal algorithm for uncalibrated cameras with different radial distortions. We show that the problem can be solved using nine points in two views by manipulating polynomials by a sequence of Gauss-Jordan eliminations in exact rational arithmetics. We demonstrate the algorithms on synthetic and real data. 1.

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Abstract. This paper presents a new fast approach to improving stability in polynomial equation solving. Gröbner basis techniques for equation solving have been applied successfully to several geometric computer vision problems. However, in many cases these methods are plagued by numerical problems. An interesting approach to stabilising the computations is to study basis selection for the quotient space C[x]/I. In this paper, the exact matrix computations involved in the solution procedure are clarified and using this knowledge we propose a new fast basis selection scheme based on QR-factorization with column pivoting. We also propose an adaptive scheme for truncation of the Gröbner basis to further improve stability. The new basis selection strategy is studied on some of the latest reported uses of Gröbner basis methods in computer vision and we demonstrate a fourfold increase in speed and nearly as good over-all precision as the previous SVD-based method. Moreover, we get typically get similar or better reduction of the largest errors. 1 1

Abstract. Finding solutions to minimal problems for estimating epipolar geometry and camera motion leads to solving systems of algebraic equations. Often, these systems are not trivial and therefore special algorithms have to be designed to achieve numerical robustness and computational efficiency. The state of the art approach for constructing such algorithms is the Gröbner basis method for solving systems of polynomial equations. Previously, the Gröbner basis solvers were designed ad hoc for concrete problems and they could not be easily applied to new problems. In this paper we propose an automatic procedure for generating Gröbner basis solvers which could be used even by non-experts to solve technical problems. The input to our solver generator is a system of polynomial equations with a finite number of solutions. The output of our solver generator is the Matlab or C code which computes solutions to this system for concrete coefficients. Generating solvers automatically opens possibilities to solve more complicated problems which could not be handled manually or solving existing problems in a better and more efficient way. We demonstrate that our automatic generator constructs efficient and numerically stable solvers which are comparable or outperform known manually constructed solvers. The automatic generator is available at

In this paper we provide new fast and simple solutions to two important minimal problems in computer vision, the five-point relative pose problem and the six-point focal length problem. We show that these two problems can easily be formulated as polynomial eigenvalue problems of degree three and two and solved using standard efficient numerical algorithms. Our solutions are somewhat more stable than state-of-the-art solutions by Nister and Stewenius and are in some sense more straightforward and easier to implement since polynomial eigenvalue problems are well studied with many efficient and robust algorithms available. The quality of the solvers is demonstrated in experiments 1. 1

Abstract—In this paper, we present three fast, hybrid numericalgebraic methods to solve polynomial systems in floating point representation, based on the eigendecomposition of a so-called multiplication matrix. In particular, these methods run using standard double precision, use only linear algebra packages, and are easy to implement. We provide the proof that these methods do indeed produce valid multiplication matrices, and show their relationship. As a specific application, we use our algorithms to compute the 3D relative translation and orientation between two robots, based on known egomotion and six robotto-robot distance measurements. Equivalently, the same system of equations arises when solving the forward kinematics of the general Stewart-Gough mechanism. Our methods can find all 40 solutions, trading off speed (0.08s to 1.5s, depending on the choice of method) for accuracy.

This paper presents a solution to the problem of pose estimation in the presence of heavy radial distortion and a potentially large number of outliers. The main contribution is an algorithm that solves for radial distortion, focal length and camera pose using a minimal set of four point correspondences between 3D world points and image points. We use a RANSAC loop to find a set of inliers and an initial estimate for bundle adjustment. Unlike previous approaches where one starts out by assuming a linear projection model, our minimal solver allows us to handle large radial distortions already at the RANSAC stage. We demonstrate that with the inclusion of radial distortion in an early stage of the process, a broader variety of cameras can be handled than was previously possible. In the experiments, no calibration whatsoever is applied to the camera. Instead we assume square pixels, zero skew and centered principal point. Although these assumptions are not strictly true, we show that good results are still obtained and by that conclude that the proposed method is applicable to uncalibrated photographs. 1.

Abstract — In this paper, we address the problem of determining the relative position and orientation (pose) of two robots navigating in 2D, based on known egomotion and noisy robot-to-robot distance measurements. We formulate this as a weighted Least Squares (WLS) estimation problem, and determine the exact global optimum by directly solving the multivariate polynomial system resulting from the first-order optimality conditions. Given the poor scalability of the original WLS problem, we propose an alternative formulation of the WLS problem in terms of squared distance measurements (squared distances WLS or SD-WLS). Using a hybrid algebraicnumeric technique, we are able to solve the corresponding firstorder optimality conditions of the SD-WLS in 125 ms in Matlab. Both methods solve the minimal (3 distance measurements) as well as the overdetermined problem (more than 3 measurements)