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.
The second recent improvement is the work of Zhang al @cite_6 . Their method works with 4 or more lines and is more accurate and robust than the method of Mirzaei and Roumeliotis. An intermediate model coordinate system is used in the method of Zhang al, which is aligned with a 3D line of longest projection. The lines are divided into triples for each of which a P3L polynomial is formed. The optimal solution of the polynomial system is selected from the roots of its derivative in terms of a least squares residual. A drawback of this algorithm is that the computational time increases strongly for higher number of lines.
@cite_6
1987385489 1546311336 2109323956 2507022162
We present a novel method for recovering the 3D-line structure of a scene from multiple widely separated views. Traditional optimization-based approaches to line-based structure from motion minimize the error between measured line segments and the projections of corresponding 3D lines. In such a case, 3D lines can be optimized using a minimum of 4 parameters. We show that this number of parameters can be further reduced by introducing additional constraints on the orientations of lines in a 3D scene. In our approach, 2D-lines are automatically detected in images with the assistance of an EM-based vanishing point estimation method which assumes the existence of edges along mutally orthogonal vanishing directions. Each detected line is automatically labeled with the orientation (e.g. vertical, horizontal) of the 3D line which generated the measurement, and it is this additional knowledge that we use to reduce the number of degrees of freedom of 3D lines during optimization. We present 3D reconstruction results for urban scenes based on manually established feature correspondences across images. We prove the approximation ratio 8 5 for the metric s,t -path-TSP, and more generally for shortest connected T-joins. The algorithm that achieves this ratio is the simple "Best of Many" version of Christofides' algorithm (1976), suggested by An, Kleinberg and Shmoys (2012), which consists in determining the best Christofides s,t -tour out of those constructed from a family @math of trees having a convex combination dominated by an optimal solution x* of the Held-Karp relaxation. They give the approximation guarantee @math for such an s,t -tour, which is the first improvement after the 5 3 guarantee of Hoogeveen's Christofides type algorithm (1991). Cheriyan, Friggstad and Gao (2012) extended this result to a 13 8-approximation of shortest connected T-joins, for |T|≥4. The ratio 8 5 is proved by simplifying and improving the approach of An, Kleinberg and Shmoys that consists in completing x* 2 in order to dominate the cost of "parity correction" for spanning trees. We partition the edge-set of each spanning tree in @math into an s,t -path (or more generally, into a T-join) and its complement, which induces a decomposition of x*. This decomposition can be refined and then efficiently used to complete x* 2 without using linear programming or particular properties of T, but by adding to each cut deficient for x* 2 an individually tailored explicitly given vector, inherent in x*. A simple example shows that the Best of Many Christofides algorithm may not find a shorter s,t -tour than 3 2 times the incidentally common optima of the problem and of its fractional relaxation. We propose a novel algorithm to register multiple 3D point sets within a common reference frame using a manifold optimization approach. The point sets are obtained with multiple laser scanners or a mobile scanner. Unlike most prior algorithms, our approach performs an explicit optimization on the manifold of rotations, allowing us to formulate the registration problem as an unconstrained minimization on a constrained manifold. This approach exploits the Lie group structure of SO 3 and the simple representation of its associated Lie algebra so 3 in terms of R3. Our contributions are threefold. We present a new analytic method based on singular value decompositions that yields a closed-form solution for simultaneous multiview registration in the noise-free scenario. Secondly, we use this method to derive a good initial estimate of a solution in the noise-free case. This initialization step may be of use in any general iterative scheme. Finally, we present an iterative scheme based on Newton's method on SO 3 that has locally quadratic convergence. We demonstrate the efficacy of our scheme on scan data taken both from the Digital Michelangelo project and from scans extracted from models, and compare it to some of the other well known schemes for multiview registration. In all cases, our algorithm converges much faster than the other approaches, (in some cases orders of magnitude faster), and generates consistently higher quality registrations. We propose a minimal solution for the similarity registration (rigid pose and scale) between two sets of 3D lines, and also between a set of co-planar points and a set of 3D lines. The first problem is solved up to 8 discrete solutions with a minimum of 2 line-line correspondences, while the second is solved up to 4 discrete solutions using 4 point-line correspondences. We use these algorithms to perform the extrinsic calibration between a pose tracking sensor and a 2D 3D ultrasound (US) curvilinear probe using a tracked needle as calibration target. The needle is tracked as a 3D line, and is scanned by the ultrasound as either a 3D line (3D US) or as a 2D point (2D US). Since the scale factor that converts US scan units to metric coordinates is unknown, the calibration is formulated as a similarity registration problem. We present results with both synthetic and real data and show that the minimum solutions outperform the correspondent non-minimal linear formulations.