Abstract This work is concerned with camera pose estimation from correspondences of 3D 2D lines, i. e. with the Perspective-n-Line (PnL) problem. We focus on large line sets, which can be efficiently solved by methods using linear formulation of PnL. We propose a novel method “DLT-Combined-Lines” based on the Direct Linear Transformation (DLT) algorithm, which benefits from a new combination of two existing DLT methods for pose estimation. The method represents 2D structure by lines, and 3D structure by both points and lines. The redundant 3D information reduces the minimum required line correspondences to 5. A cornerstone of the method is a combined projection matrix estimated by the DLT algorithm. It contains multiple estimates of camera rotation and translation, which can be recovered after enforcing constraints of the matrix. Multiplicity of the estimates is exploited to improve the accuracy of the proposed method. For large line sets (10 and more), the method is comparable to the state-of-the-art in accuracy of orientation estimation. It achieves state-of-the-art accuracy in estimation of camera position and it yields the smallest reprojection error under strong image noise. The method achieves top-3 results on real world data. The proposed method is also highly computationally effective, estimating the pose of 1000 lines in 12 ms on a desktop computer.
The second recent improvement is the Robust PnL () algorithm of @cite_11 . 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 , 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.
@cite_11
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We propose a non-iterative solution for the Perspective-n-Line (PnL) problem, which can efficiently and accurately estimate the camera pose for both small number and large number of line correspondences. By selecting a rotation axis in the camera framework, the reference lines are divided into triplets to form a sixteenth order cost function, and then the optimum is retrieved from the roots of the derivative of the cost function by evaluating the orthogonal errors and the reprojected errors of the local minima. The final pose estimation is normalized by a 3D alignment approach. The advantages of the proposed method are as follows: (1) it stably retrieves the optimum of the solution with very little computational complexity and high accuracy; (2) small line sets can be robustly handled to achieve highly accurate results and; (3) large line sets can be efficiently handled because it is O(n). 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. 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 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 address the problem of camera motion and 3D structure reconstruction from line correspondences across multiple views, from initialization to final bundle adjustment. One of the main difficulties when dealing with line features is their algebraic representation. First, we consider the triangulation problem. Based on Plucker coordinates to represent the 3D lines, we propose a maximum likelihood algorithm, relying on linearizing the Plucker constraint and on a Plucker correction procedure, computing the closest Plucker coordinates to a given 6-vector. Second, we consider the bundle adjustment problem, which is essentially a nonlinear optimization process on camera motion and 3D line parameters. Previous overparameterizations of 3D lines induce gauge freedoms and or internal consistency constraints. We propose the orthonormal representation, which allows handy nonlinear optimization of 3D lines using the minimum four parameters with an unconstrained optimization engine. We compare our algorithms to existing ones on simulated and real data. Results show that our triangulation algorithm outperforms standard linear and bias-corrected quasi-linear algorithms, and that bundle adjustment using our orthonormal representation yields results similar to the standard maximum likelihood trifocal tensor algorithm, while being usable for any number of views. 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). We present an autocalibration algorithm for upgrading a projective reconstruction to a metric reconstruction by estimating the absolute dual quadric. The algorithm enforces the rank degeneracy and the positive semidefiniteness of the dual quadric as part of the estimation procedure, rather than as a post-processing step. Furthermore, the method allows the user, if he or she so desires, to enforce conditions on the plane at infinity so that the reconstruction satisfies the chirality constraints. The algorithm works by constructing low degree polynomial optimization problems, which are solved to their global optimum using a series of convex linear matrix inequality relaxations. The algorithm is fast, stable, robust and has time complexity independent of the number of views. We show extensive results on synthetic as well as real datasets to validate our algorithm. In this paper, we propose an efficient algorithm, called vector field consensus, for establishing robust point correspondences between two sets of points. Our algorithm starts by creating a set of putative correspondences which can contain a very large number of false correspondences, or outliers, in addition to a limited number of true correspondences (inliers). Next, we solve for correspondence by interpolating a vector field between the two point sets, which involves estimating a consensus of inlier points whose matching follows a nonparametric geometrical constraint. We formulate this a maximum a posteriori (MAP) estimation of a Bayesian model with hidden latent variables indicating whether matches in the putative set are outliers or inliers. We impose nonparametric geometrical constraints on the correspondence, as a prior distribution, using Tikhonov regularizers in a reproducing kernel Hilbert space. MAP estimation is performed by the EM algorithm which by also estimating the variance of the prior model (initialized to a large value) is able to obtain good estimates very quickly (e.g., avoiding many of the local minima inherent in this formulation). We illustrate this method on data sets in 2D and 3D and demonstrate that it is robust to a very large number of outliers (even up to 90 ). We also show that in the special case where there is an underlying parametric geometrical model (e.g., the epipolar line constraint) that we obtain better results than standard alternatives like RANSAC if a large number of outliers are present. This suggests a two-stage strategy, where we use our nonparametric model to reduce the size of the putative set and then apply a parametric variant of our approach to estimate the geometric parameters. Our algorithm is computationally efficient and we provide code for others to use it. In addition, our approach is general and can be applied to other problems, such as learning with a badly corrupted training data set. In this article, we introduce a new global optimization method to the field of multiview 3D reconstruction. While global minimization has been proposed in a discrete formulation in form of the maxflow-mincut framework, we suggest the use of a continuous convex relaxation scheme. Specifically, we propose to cast the problem of 3D shape reconstruction as one of minimizing a spatially continuous convex functional. In qualitative and quantitative evaluation we demonstrate several advantages of the proposed continuous formulation over the discrete graph cut solution. Firstly, geometric properties such as weighted boundary length and surface area are represented in a numerically consistent manner: The continuous convex relaxation assures that the algorithm does not suffer from metrication errors in the sense that the reconstruction converges to the continuous solution as the spatial resolution is increased. Moreover, memory requirements are reduced, allowing for globally optimal reconstructions at higher resolutions. We study three different energy models for multiview reconstruction, which are based on a common variational template unifying regional volumetric terms and on-surface photoconsistency. The three models use data measurements at increasing levels of sophistication. While the first two approaches are based on a classical silhouette-based volume subdivision, the third one relies on stereo information to define regional costs. Furthermore, this scheme is exploited to compute a precise photoconsistency measure as opposed to the classical estimation. All three models are compared on standard data sets demonstrating their advantages and shortcomings. For the third one, which gives the most accurate results, a more exhaustive qualitative and quantitative evaluation is presented. 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. We propose a noniterative solution for the Perspective-n-Point ( P n P ) problem, which can robustly retrieve the optimum by solving a seventh order polynomial. The central idea consists of three steps: 1) to divide the reference points into 3-point subsets in order to achieve a series of fourth order polynomials, 2) to compute the sum of the square of the polynomials so as to form a cost function, and 3) to find the roots of the derivative of the cost function in order to determine the optimum. The advantages of the proposed method are as follows: First, it can stably deal with the planar case, ordinary 3D case, and quasi-singular case, and it is as accurate as the state-of-the-art iterative algorithms with much less computational time. Second, it is the first noniterative P n P solution that can achieve more accurate results than the iterative algorithms when no redundant reference points can be used (n 5). Third, large-size point sets can be handled efficiently because its computational complexity is O(n). 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. Abstract This paper mathematically analyzes and proposes new solutions for the problem of estimating the camera 3D location and orientation ( pose determination ) from a matched set of 3D model and 2D image landmark features. Least-squares techniques for line tokens, which minimize both rotation and translation simultaneously, are developed and shown to be far superior to the earlier techniques which solved for rotation first and then translation. However, least-squares techniques fail catastrophically when outliers (or gross errors) are present in the match data. Outliers arise frequently due to incorrect correspondences or gross errors in the 3D model. Robust techniques for pose determination are developed to handle data contaminated by fewer than 50.0 outliers. Finally, the sensitivity of pose determination to incorrect estimates of camera parameters is analyzed. It is shown that for small field of view systems, offsets in the image center do not significantly affect the location of the camera in a world coordinate system. Errors in the focal length significantly affect only the component of translation along the optical axis in the pose computation. 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. This paper introduces an algorithm for the registration of rotated and translated volumes using the three-dimensional (3-D) pseudopolar Fourier transform, which accurately computes the Fourier transform of the registered volumes on a near-spherical 3-D domain without using interpolation. We propose a three-step procedure. The first step estimates the rotation axis. The second step computes the planar rotation relative to the rotation axis. The third step recovers the translational displacement. The rotation estimation is based on Euler's theorem, which allows one to represent a 3-D rotation as a planar rotation around a 3-D rotation axis. This axis is accurately recovered by the 3-D pseudopolar Fourier transform using radial integrations. The residual planar rotation is computed by an extension of the angular difference function to cylindrical motion. Experimental results show that the algorithm is accurate and robust to noise In this paper, we consider conic programming problems whose constraints consist of linear equalities, linear inequalities, a nonpolyhedral cone, and a polyhedral cone. A convenient way for solving this class of problems is to apply the directly extended alternating direction method of multipliers (ADMM) to its dual problem, which has been observed to perform well in numerical computations but may diverge in theory. Ideally, one should find a convergent variant which is at least as efficient as the directly extended ADMM in practice. We achieve this goal by designing a convergent semiproximal ADMM (called sPADMM3c for convenience) for convex programming problems having three separable blocks in the objective function with the third part being linear. At each iteration, the proposed sPADMM3c takes one special block coordinate descent (BCD) cycle with the order @math , instead of the usual @math Gauss--Seidel BCD cycle used in the nonconvergent directly extended 3-block ADMM, for updating the variable blocks. Our numerical experiments demonstrate that the convergent method is at least 20 faster than the directly extended ADMM with unit step-length for the vast majority of about 550 large-scale doubly nonnegative semidefinite programming problems with linear equality and or inequality constraints. This confirms that at least for conic convex programming, one can design a convergent and efficient ADMM with a special BCD cycle of updating the variable blocks. Abstract This paper introduces a new method of registering point sets. The registration error is directly minimized using general-purpose non-linear optimization (the Levenberg–Marquardt algorithm). The surprising conclusion of the paper is that this technique is comparable in speed to the special-purpose Iterated Closest Point algorithm, which is most commonly used for this task. Because the routine directly minimizes an energy function, it is easy to extend it to incorporate robust estimation via a Huber kernel, yielding a basin of convergence that is many times wider than existing techniques. Finally, we introduce a data structure for the minimization based on the chamfer distance transform, which yields an algorithm that is both faster and more robust than previously described methods. This paper presents a new approach to the multi-robot map-alignment problem that enables teams of robots to build joint maps without initial knowledge of their relative poses. The key contribution of this work is an optimal algorithm for merging (not necessarily overlapping) maps that are created by different robots independently. Relative pose measurements between pairs of robots are processed to compute the coordinate transformation between any two maps. Noise in the robot-to-robot observations, propagated through the map-alignment process, increases the error in the position estimates of the transformed landmarks, and reduces the overall accuracy of the merged map. When there is overlap between the two maps, landmarks that appear twice provide additional information, in the form of constraints, which increases the alignment accuracy. Landmark duplicates are identified through a fast nearest-neighbor matching algorithm. In order to reduce the computational complexity of this search process, a kd-tree is used to represent the landmarks in the original map. The criterion employed for matching any two landmarks is the Mahalanobis distance. As a means of validation, we present experimental results obtained from two robots mapping an area of 4,800 m2 The Perspective-Three-Point (P3P) problem aims at determining the position and orientation of the camera in the world reference frame from three 2D-3D point correspondences. This problem is known to provide up to four solutions that can then be disambiguated using a fourth point. All existing solutions attempt to first solve for the position of the points in the camera reference frame, and then compute the position and orientation of the camera in the world frame, which alignes the two point sets. In contrast, in this paper we propose a novel closed-form solution to the P3P problem, which computes the aligning transformation directly in a single stage, without the intermediate derivation of the points in the camera frame. This is made possible by introducing intermediate camera and world reference frames, and expressing their relative position and orientation using only two parameters. The projection of a world point into the parametrized camera pose then leads to two conditions and finally a quartic equation for finding up to four solutions for the parameter pair. A subsequent backsubstitution directly leads to the corresponding camera poses with respect to the world reference frame. We show that the proposed algorithm offers accuracy and precision comparable to a popular, standard, state-of-the-art approach but at much lower computational cost (15 times faster). Furthermore, it provides improved numerical stability and is less affected by degenerate configurations of the selected world points. The superior computational efficiency is particularly suitable for any RANSAC-outlier-rejection step, which is always recommended before applying PnP or non-linear optimization of the final solution. Reproducible positioning of the patient during fractionated external beam radiation therapy is imperative to ensure that the delivered dose distribution matches the planned one. In this paper, we expand on a 2D-3D image registration method to verify a patient’s setup in three dimensions (rotations and translations) using orthogonal portal images and megavoltage digitally reconstructedradiographs (MDRRs) derived from CT data. The accuracy of 2D-3D registration was improved by employing additional image preprocessing steps and a parabolic fit to interpolate the parameter space of the cost function utilized for registration. Using a humanoid phantom, precision for registration of three-dimensional translations was found to be better than 0.5 mm (1 s.d.) for any axis when no rotations were present. Three-dimensional rotations about any axis were registered with a precision of better than 0.2 ° (1 s.d.) when no translations were present. Combined rotations and translations of up to 4 ° and 15 mm were registered with 0.4 ° and 0.7 mm accuracy for each axis. The influence of setup translations on registration of rotations and vice versa was also investigated and mostly agrees with a simple geometric model. Additionally, the dependence of registration accuracy on three cost functions, angular spacing between MDRRs, pixel size, and field-of-view, was examined. Best results were achieved by mutual information using 0.5 ° angular spacing and a 10 × 10 cm 2 field-of-view with 140 × 140 pixels . Approximating patient motion as rigid transformation, the registration method is applied to two treatment plans and the patients’ setup errors are determined. Their magnitude was found to be ⩽ 6.1 mm and ⩽ 2.7 ° for any axis in all of the six fractions measured for each treatment plan.