Correspondences between 3D lines and their 2D images captured by a camera are often used to determine position and orientation of the camera in space. In this work, we propose a novel algebraic algorithm to estimate the camera pose. We parameterize 3D lines using Plucker coordinates that allow linear projecti on of the lines into the image. A line projection matrix is estimated using Linear Least Squares and the camera pose is then extracted from the matrix. An algebraic approach to handle mismatched line correspondences is also included. The proposed algorithm is an order of magnitude faster yet comparably accurate and robust to the state-of-the-art, it does not require initialization, and it yields only one solution. The described method requires at least 9 lines and is particularly suitable for scenarios with 25 and more lines, as also shown in the results.
Recently, two major improvements of algebraic approaches have been achieved. First, Mirzaei and Roumeliotis @cite_4 proposed a method which is both efficient (linear computational complexity depending on the number of lines) and robust in the presence of image noise. The cases with 3 or more lines can be handled. A polynomial system with 27 candidate solutions is constructed and solved through the eigendecomposition of a multiplication matrix. Camera orientations having the least square error are considered to be the optimal ones. Camera positions are obtained separately using the Linear Least Squares. Nonetheless, the problem of this algorithm is that it often yields multiple solutions.
@cite_4
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Correspondences between 2D lines in an image and 3D lines in the surrounding environment can be exploited to determine the camera's position and attitude (pose). In this paper, we introduce a novel approach to estimate the camera's pose by directly solving the corresponding least-squares problem algebraically. Specifically, the optimality conditions of the least-squares problem form a system of polynomial equations, which we efficiently solve through the eigendecomposition of a so-called multiplication matrix. Contrary to existing methods, the proposed algorithm (i) is guaranteed to find the globally optimal estimate in the least-squares sense, (ii) does not require initialization, and (iii) has computational cost only linear in the number of measurements. The superior performance of the proposed algorithm compared to previous approaches is demonstrated through extensive simulations and experiments. We describe two direct quasilinear methods for camera pose (absolute orientation) and calibration from a single image of 4 or 5 known 3D points. They generalize the 6 point 'Direct Linear Transform' method by incorporating partial prior camera knowledge, while still allowing some unknown calibration parameters to be recovered. Only linear algebra is required, the solution is unique in non-degenerate cases, and additional points can be included for improved stability. Both methods fail for coplanar points, but we give an experimental eigendecomposition based one that handles both planar and nonplanar cases. Our methods use recent polynomial solving technology, and we give a brief summary of this. One of our aims was to try to understand the numerical behaviour of modern polynomial solvers on some relatively simple test cases, with a view to other vision applications. In this work, we present a Direct Least-Squares (DLS) method for computing all solutions of the perspective-n-point camera pose determination (PnP) problem in the general case (n ≥ 3). Specifically, based on the camera measurement equations, we formulate a nonlinear least-squares cost function whose optimality conditions constitute a system of three third-order polynomials. Subsequently, we employ the multiplication matrix to determine all the roots of the system analytically, and hence all minima of the LS, without requiring iterations or an initial guess of the parameters. A key advantage of our method is scalability, since the order of the polynomial system that we solve is independent of the number of points. We compare the performance of our algorithm with the leading PnP approaches, both in simulation and experimentally, and demonstrate that DLS consistently achieves accuracy close to the Maximum-Likelihood Estimator (MLE). A method is presented for building solvers for classes of multivariate polynomial equations. The method is based on solving an analogous template problem over a finite field, and then using the elimination order established for this problem for the original class of problems. A strength of this method is that this permits pivoting in the elimination. Solvers for several minimal problems in computer vision are presented. Relative pose is solved both for a generalised camera, and for a camera with unknown focal length, both in two positions with six visible points. A solver for optimal triangulation in three images is presented. Model-free calibration for pinhole cameras is investigated. It is shown that for a smooth deformation of the image plane, the image plane can be projectively reconstructed from two flow-fields from purely translating cameras. Methods for hand-eye calibration using the multilinear constraints and vehicle-eye for laser-scanner based navigation systems are presented. (Less)