Newton-Type Iterative Solver for Multiple View L2L2 Triangulation

In this note, we show that the L2 optimal solutions to most real multiple view L2 triangulation problems can be efficiently obtained by two-stage Newton-like iterative methods, while the difficulty of such problems mainly lies in how to verify the L2 optimality. Such a working two-stage bundle adjustment approach features: first, the algorithm is initialized by symmedian point triangulation, a multiple-view generalization of the mid-point method; second, a symbolic-numeric method is employed to compute derivatives accurately; third, globalizing strategy such as line search or trust region is smoothly applied to the underlying iteration which assures algorithm robustness in general cases. Numerical comparison with tfml method shows that the local minimizers obtained by the two-stage iterative bundle adjustment approach proposed here are also the L2 optimal solutions to all the calibrated data sets available online by the Oxford visual geometry group. Extensive numerical experiments indicate the bundle adjustment approach solves more than 99% the real triangulation problems optimally. An IEEE 754 double precision C++ implementation shows that it takes only about 0.205 second tocompute allthe 4983 points in the Oxford dinosaur data setvia Gauss-Newton iteration hybrid with a line search strategy on a computer with a 3.4GHz Intel i7 CPU.Comment: 15 pages, 1 figure, 4 tables, 30 references, C++ source code

Fundamental Matrix Estimation: A Study of Error Criteria

The fundamental matrix (FM) describes the geometric relations that exist between two images of the same scene. Different error criteria are used for estimating FMs from an input set of correspondences. In this paper, the accuracy and efficiency aspects of the different error criteria were studied. We mathematically and experimentally proved that the most popular error criterion, the symmetric epipolar distance, is biased. It was also shown that despite the similarity between the algebraic expressions of the symmetric epipolar distance and Sampson distance, they have different accuracy properties. In addition, a new error criterion, Kanatani distance, was proposed and was proved to be the most effective for use during the outlier removal phase from accuracy and efficiency perspectives. To thoroughly test the accuracy of the different error criteria, we proposed a randomized algorithm for Reprojection Error-based Correspondence Generation (RE-CG). As input, RE-CG takes an FM and a desired reprojection error value $d$. As output, RE-CG generates a random correspondence having that error value. Mathematical analysis of this algorithm revealed that the success probability for any given trial is 1 - (2/3)^2 at best and is 1 - (6/7)^2 at worst while experiments demonstrated that the algorithm often succeeds after only one trial.Comment: 15 pages, 7 figures, Pattern Recognition Letters, 201

Hand-guided 3D surface acquisition by combining simple light sectioning with real-time algorithms

Precise 3D measurements of rigid surfaces are desired in many fields of application like quality control or surgery. Often, views from all around the object have to be acquired for a full 3D description of the object surface. We present a sensor principle called "Flying Triangulation" which avoids an elaborate "stop-and-go" procedure. It combines a low-cost classical light-section sensor with an algorithmic pipeline. A hand-guided sensor captures a continuous movie of 3D views while being moved around the object. The views are automatically aligned and the acquired 3D model is displayed in real time. In contrast to most existing sensors no bandwidth is wasted for spatial or temporal encoding of the projected lines. Nor is an expensive color camera necessary for 3D acquisition. The achievable measurement uncertainty and lateral resolution of the generated 3D data is merely limited by physics. An alternating projection of vertical and horizontal lines guarantees the existence of corresponding points in successive 3D views. This enables a precise registration without surface interpolation. For registration, a variant of the iterative closest point algorithm - adapted to the specific nature of our 3D views - is introduced. Furthermore, data reduction and smoothing without losing lateral resolution as well as the acquisition and mapping of a color texture is presented. The precision and applicability of the sensor is demonstrated by simulation and measurement results.Comment: 19 pages, 22 figure

Parallel and Scalable Heat Methods for Geodesic Distance Computation

In this paper, we propose a parallel and scalable approach for geodesic distance computation on triangle meshes. Our key observation is that the recovery of geodesic distance with the heat method from [Crane et al. 2013] can be reformulated as optimization of its gradients subject to integrability, which can be solved using an efficient first-order method that requires no linear system solving and converges quickly. Afterward, the geodesic distance is efficiently recovered by parallel integration of the optimized gradients in breadth-first order. Moreover, we employ a similar breadth-first strategy to derive a parallel Gauss-Seidel solver for the diffusion step in the heat method. To further lower the memory consumption from gradient optimization on faces, we also propose a formulation that optimizes the projected gradients on edges, which reduces the memory footprint by about 50%. Our approach is trivially parallelizable, with a low memory footprint that grows linearly with respect to the model size. This makes it particularly suitable for handling large models. Experimental results show that it can efficiently compute geodesic distance on meshes with more than 200 million vertices on a desktop PC with 128GB RAM, outperforming the original heat method and other state-of-the-art geodesic distance solvers

Complete End-To-End Low Cost Solution To a 3D Scanning System with Integrated Turntable

3D reconstruction is a technique used in computer vision which has a wide range of applications in areas like object recognition, city modelling, virtual reality, physical simulations, video games and special effects. Previously, to perform a 3D reconstruction, specialized hardwares were required. Such systems were often very expensive and was only available for industrial or research purpose. With the rise of the availability of high-quality low cost 3D sensors, it is now possible to design inexpensive complete 3D scanning systems. The objective of this work was to design an acquisition and processing system that can perform 3D scanning and reconstruction of objects seamlessly. In addition, the goal of this work also included making the 3D scanning process fully automated by building and integrating a turntable alongside the software. This means the user can perform a full 3D scan only by a press of a few buttons from our dedicated graphical user interface. Three main steps were followed to go from acquisition of point clouds to the finished reconstructed 3D model. First, our system acquires point cloud data of a person/object using inexpensive camera sensor. Second, align and convert the acquired point cloud data into a watertight mesh of good quality. Third, export the reconstructed model to a 3D printer to obtain a proper 3D print of the model.Comment: 22 page

Adaptive Hybridizable Discontinuous Galerkin discretization of the Grad-Shafranov equation by extension from polygonal subdomains

We propose a high-order adaptive numerical solver for the semilinear elliptic boundary value problem modelling magnetic plasma equilibrium in axisymmetric confinement devices. In the fixed boundary case, the equation is posed on curved domains with piecewise smooth curved boundaries that may present corners. The solution method we present is based on the hybridizable discontinuous Galerkin method and sidesteps the need for geometry-conforming triangulations thanks to a transfer technique that allows to approximate the solution using only a polygonal subset as computational domain. Moreover, the solver features automatic mesh refinement driven by a residual-based a posteriori error estimator. As the mesh is locally refined, the computational domain is automatically updated in order to always maintain the distance between the actual boundary and the computational boundary of the order of the local mesh diameter. Numerical evidence is presented of the suitability of the estimator as an approximate error measure for physically relevant equilibria with pressure pedestals, internal transport barriers, and current holes on realistic geometries

Localization in internets of mobile agents: A linear approach

Fifth generation~(5G) networks providing much higher bandwidth and faster data rates will allow connecting vast number of static and mobile devices, sensors, agents, users, machines, and vehicles, supporting Internet-of-Things (IoT), real-time dynamic networks of mobile things. Positioning and location awareness will become increasingly important, enabling deployment of new services and contributing to significantly improving the overall performance of the 5G~system. Many of the currently talked about solutions to positioning in~5G are centralized, mostly requiring direct line-of-sight (LoS) to deployed access nodes or anchors at the same time, which in turn requires high-density deployments of anchors. But these LoS and centralized positioning solutions may become unwieldy as the number of users and devices continues to grow without limit in sight. As an alternative to the centralized solutions, this paper discusses distributed localization in a 5G enabled IoT environment where many low power devices, users, or agents are to locate themselves without global or LoS access to anchors. Even though positioning is essentially a non-linear problem (solving circle equations by trilateration or triangulation), we discuss a cooperative \textit{linear} distributed iterative solution with only local measurements, communication and computation needed at each agent. Linearity is obtained by reparametrization of the agent location through barycentric coordinate representations based on local neighborhood geometry that may be computed in terms of certain Cayley-Menger determinants involving relative local inter-agent distance measurements.Comment: Technical Repor

Closed-Form Optimal Two-View Triangulation Based on Angular Errors

In this paper, we study closed-form optimal solutions to two-view triangulation with known internal calibration and pose. By formulating the triangulation problem as $L_1$ and $L_\infty$ minimization of angular reprojection errors, we derive the exact closed-form solutions that guarantee global optimality under respective cost functions. To the best of our knowledge, we are the first to present such solutions. Since the angular error is rotationally invariant, our solutions can be applied for any type of central cameras, be it perspective, fisheye or omnidirectional. Our methods also require significantly less computation than the existing optimal methods. Experimental results on synthetic and real datasets validate our theoretical derivations.Comment: Accepted to ICCV201

Generalised primal-dual grids for unstructured co-volume schemes

The generation of high-quality staggered unstructured grids is considered, leading to the development of a new optimisation-based strategy designed to construct weighted `Regular-Power' tessellations appropriate for co-volume type numerical discretisation techniques. This new framework aims to extend the conventional Delaunay-Voronoi primal-dual structure; seeking to assemble generalised orthogonal tessellations with enhanced geometric quality. The construction of these grids is motivated by the desire to improve the performance and accuracy of numerical methods based on unstructured co-volume type schemes, including various staggered grid techniques for the simulation of fluid dynamics and hyperbolic transport. In this study, a new hybrid optimisation strategy is proposed; seeking to optimise the geometry, topology and weights associated with general, two-dimensional Regular-Power tessellations using a combination of gradient-ascent and energy-based techniques. The performance of this new method is tested experimentally, with a range of complex, multi-resolution primal-dual grids generated for various coastal and regional ocean modelling applications

An iterative domain decomposition method for free boundary problems with nonlinear flux jump constraint

In this paper we design an iterative domain decomposition method for free boundary problems with nonlinear flux jump condition. Our approach is related to damped Newton's methods. The proposed scheme requires, in each iteration, the approximation of the flux on (both sides of) the free interface. We present a Finite Element implementation of our method. The numerical implementation uses harmonically deformed triangulations to inexpensively generate finite element meshes in subdomains. We apply our method to a simplified model for jet flows in pipes and to a simple magnetohydrodynamics model. Finally, we present numerical examples studying the convergence of our scheme