Epipolar geometry and relative camera pose computation are examples of tasks which can be formulated as minimal problems and solved from a minimal number of image points. Finding the solution 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. In this paper we provide a solution to the problem of estimating radial distortion and epipolar geometry from eight correspondences in two images. Unlike previous algorithms, which were able to solve the problem from nine correspondences only, we enforce the determinant of the fundamental matrix be zero. This leads to a system of eight quadratic and one cubic equation in nine variables. We simplify the system by eliminating six of these variables. Then, we solve the system by finding eigenvectors of an action matrix of a suitably chosen polynomial. We show how to construct the action matrix without computing complete Gröbner basis, which provides an efficient and robust solver. The quality of the solver is demonstrated on synthetic and real data. 1.

We propose a non-iterative solution to the PnP problem—the estimation of the pose of a calibrated camera from n 3D-to-2D point correspondences—whose computational complexity grows linearly with n. This is in contrast to state-of-the-art methods that are O(n 5) or even O(n 8), without being more accurate. Our method is applicable for all n ≥ 4 and handles properly both planar and non-planar configurations. Our central idea is to express the n 3D points as a weighted sum of four virtual control points. The problem then reduces to estimating the coordinates of these control points in the camera referential, which can be done in O(n) time by expressing these coordinates as weighted sum of the eigenvectors of a 12 × 12 matrix and solving a small constant number of quadratic equations to pick the right weights. The advantages of our method are demonstrated by thorough testing on both synthetic and real-data.

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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Abstract — We present a system for Monocular Simultaneous Localization and Mapping (Mono-SLAM) relying solely on video input. Our algorithm makes it possible to precisely estimate the camera trajectory without relying on any motion model. The estimation is fully incremental: at a given time frame, only the current location is estimated while the previous camera positions are never modified. In particular, we do not perform any simultaneous iterative optimization of the camera positions and estimated 3D structure (local bundle adjustment). The key aspects of the system is a fast and simple pose estimation algorithm that uses information not only from the estimated 3D map, but also from the epipolar constraint. We show that the latter leads to a much more stable estimation of the camera trajectory than the conventional approach. We perform high precision camera trajectory estimation in urban scenes with a large amount of clutter. Using an omnidirectional camera placed on a vehicle, we cover the longest distance ever reported, up to 2.5 kilometers. I.

We propose a non-iterative solution to the PnP problem—the estimation of the pose of a calibrated camera from n 3D-to-2D point correspondences—whose computational complexity grows linearly with n. This is in contrast to state-of-the-art methods that are O(n 5) or even O(n 8), without being more accurate. Our method is applicable for all n ≥ 4 and handles properly both planar and non-planar configurations. Our central idea is to express the n 3D points as a weighted sum of four virtual control points. The problem then reduces to estimating the coordinates of these control points in the camera referential, which can be done in O(n) time by expressing these coordinates as weighted sum of the eigenvectors of a 12 × 12 matrix and solving a small constant number of quadratic equations to pick the right weights. Furthermore, if maximal precision is required, the output of the closed-form solution can be used to initialize a Gauss-Newton scheme, which improves accuracy with negligible amount of additional time. The advantages of our method are demonstrated by thorough testing on both synthetic and real-data.

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. 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