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We survey 36 curve reconstruction algorithms and compare 14 of these with quantitative and qualitative analysis. As inputs, we take unorganized points, samples on the boundary of binary images or smooth curves, and evaluate with ground truth.
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Curve reconstruction from unstructured points in a plane is a fundamental problem with many applications that has generated research interest for decades. Involved aspects like handling open, sharp, multiple and non‐manifold outlines, run‐time and provability as well as potential extension to 3D for surface reconstruction have led to many different...
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... In cases, where a sufficient number points in the point-cloud data cannot be guaranteed to obtain a highly accurate description of the complex geometry of the structure under investigation, the convergence curve will stagnate after a certain number of mesh refinement steps as the geometric error dominates the overall behavior of the method, as reported by Kudela et al. [14]. In addition, it is worth mentioning that Ohrhallinger et al. [86] conducted a thorough investigation on the consequences of different geometric reconstruction schemes using discrete points in 2D. A variety of ill-conditioned geometries including noisy data, sharp features, and non-manifold curves have been reported. ...
Point-cloud-based geometry acquisition techniques have seen an increased deployment in recent years and can be used in many branches of science and engineering. Different from conventional computer aided design (CAD) models, point-cloud data can be directly obtained by applying surveying technology, such as laser scanning devices. This opens a pathway to an efficient and highly automatic acquisition of the geometric model. However, in order to perform numerical analysis on point-cloud models using conventional finite element approaches, a surface reconstruction is required, which is both time consuming and error-prone. Therefore, a direct numerical analysis framework based on point-cloud data in conjunction with polyhedral element techniques is developed in this contribution. To this end, a robust and efficient meshing paradigm based on the direct use of point-cloud data (without surface reconstruction) is proposed. From the geometric model, a trimmed octree mesh containing only a limited number of master elements (element patterns) is directly generated, which is highly suitable for pre-computation techniques and parallelization (high-performance computing-HPC). By computing the element stiffness and mass matrices from pre-computed master elements via scaling, a fully automatic HPC framework is developed. By means of six numerical examples, we demonstrate the robustness, versatility, and efficiency of the developed methodology in handling (geometrically) complex engineering problems.
... In the context of comparing our algorithm with existing state-of-the-art methods for contour reconstruction from 2D point sets, we primarily focused on analyzing two algorithms: HNN-Crust [28] and Vicur [24]. Our selection of these two was informed by a thorough review paper [11], where they particularly distinguished themselves from other methods in an experimental study. We also selected these two algorithms because, as highlighted in Section 2, they represent cutting-edge techniques within their respective algorithm categories. ...
Realistic sensory feedback is paramount for amputees as it improves prosthetic limb control and boosts functionality, safety, and overall quality of life. This sensory restoration relies on the direct electrostimulation of residual peripheral nerves. Computational models are instrumental in simulating these neurostimulation effects, offering solutions to the complexities tied to extensive animal/human trials and costly materials. Central to these models is the detailed mapping of nerve geometry, necessitating the delineation of internal nerve structures, such as fascicles, across various cross-sections. In our modeling process, we faced the challenge of organizing an originally unstructured set of points into coherent contours. We introduced a parameter-free curve-reconstruction algorithm that combines valley-seeking clustering, an adaptive Kalman filter, and the nearest neighbor classification technique. While intuitively simple for humans, the task of reconstructing multiple open and/or closed lines with pronounced corners from a nonuniform point set is daunting for many algorithms. Additionally, the precise differentiation of adjacent curves, commonly encountered in realistic nerve models, remains a formidable challenge even for top-tier algorithms. Our proposed method adeptly navigates the complexities inherent to nerve structure reconstruction. While our algorithm is chiefly designed for closed curves, as dictated by nerve geometry, we believe it can be reconfigured with appropriate code adjustments to handle open curves. Beyond neuroprosthetics, our proposed model has the potential to be applied and spark innovations in biomedicine and a variety of other fields.
... Previous approaches that were used to generate curves from points include Crust [14], Connect-2d [15], Crawl, Peel, Gathan, nn-Crust [16], and SIGDT [17]. These are known as curve reconstruction (CR) methods. ...
... Existing CR algorithms such as Crust [14], Connect-2d [15], Crawl, Peel, Gathan, nn-Crust [16], and SIGDT [17] do not perform well on sparse data. These methods are based on proximity and continuity measures. ...
... In cases, where a sufficient number points in the point-cloud data cannot be guaranteed to obtain a highly accurate description of the complex geometry of the structure under investigation, the convergence curve will stagnate after a certain number of mesh refinement steps as the geometric error dominates the overall behavior of the method, as reported by Kudela et al. [14]. In addition, it is worth mentioning that Ohrhallinger et al. [86] conducted a thorough investigation on the consequences of different geometric reconstruction schemes using discrete points in 2D. A variety of ill-conditioned geometries including noisy data, sharp features, and non-manifold curves have been reported. ...
... In the literature, there are numerous methods for reconstructing unorganized points. Ohrhallinger et al. [9] provide an overview and comparison of existing algorithms, including the approach presented in [10], which uses a minimum spanning tree (MST) for curve reconstruction. A minimum spanning tree is a cycle-free graph that connects all points while minimizing the edge length. ...
Many applications today require the precise determination of the position and orientation of a moving platform over time. However, especially in safety-critical areas, it is also important to derive quality characteristics of the trajectory estimation. This allows verification that sensors are operating within the precision and accuracy required for the application. In this paper, we propose a methodology for trajectory evaluation and address the challenges involved. Our approach is based on repeated measurements obtained using a closed loop rail track and allows the evaluation of the trajectory estimation in terms of precision and accuracy. Starting with the chronologically ordered raw data, the methodology first spatially sorts the measurements and then approximates them to a mean trajectory. The deviations between the single pose observations and the mean trajectory indicate the precision of the observed poses. With the addition of a higher-order reference, our methodology also determines the accuracy of the system under test. The applicability of our method is demonstrated by an exemplary evaluation of a low-cost inertial navigation system.
... However, a more detailed study is beyond the scope of this paper. Instead we would like to point out that 2D point cloud curve reconstruction is indeed an active field of research and direct the interested reader to the very recent state of the art review published in [32]. ...
... .de /cie _sam _public /fcmlab/ In section 3.2 we then apply the sharp interface approach on two point cloud examples taken from a benchmark-series of point clouds [32] which was built for the evaluation of reconstruction algorithms to demonstrate that the algorithm is capable of capturing also examples with convex, concave and non-smoothly boundaries involving kinks. ...
... The point clouds in question are taken from the benchmark by Ohrhallinger et al. [32] and may be found at https://gitlab .com / stefango74 /curve -benchmark. ...
This paper develops and investigates a new method for the application of Dirichlet boundary conditions for computational models defined by point clouds. Point cloud models often stem from laser or structured-light scanners which are used to scan existing mechanical structures for which CAD models either do not exist or from which the artifact under investigation deviates in shape or topology. Instead of reconstructing a CAD model from point clouds via surface reconstruction and a subsequent boundary conforming mesh generation, a direct analysis without pre-processing is possible using embedded domain finite element methods. These methods use non-boundary conforming meshes which calls for a weak enforcement of Dirichlet boundary conditions. For point cloud based models, Dirichlet boundary conditions are usually imposed using a diffuse interface approach. This leads to a significant computational overhead due to the necessary computation of domain integrals. Additionally, undesired side effects on the gradients of the solution arise which can only be controlled to some extent. This paper develops a new sharp interface approach for point cloud based models which avoids both issues. The computation of domain integrals is circumvented by an implicit approximation of corresponding Voronoi diagrams of higher order and the resulting sharp approximation avoids the side effects of diffuse approaches. Benchmark examples from the graphics as well as the computational mechanics community are used to verify the algorithm. All algorithms are implemented in the FCMLab framework and provided at https://gitlab.lrz.de/cie_sam_public/fcmlab/. Further, we discuss challenges and limitations of point cloud based analysis w.r.t. application of Dirichlet boundary conditions.
... However, a more detailed study is beyond the scope of this paper. Instead we would like to point out that 2D point cloud curve reconstruction is indeed an active field of research and direct the interested reader to the very recent state of the art review published in [31]. ...
... All algorithms are implemented in the FCMLab framework which documented in [32] and publicly accessible at https://gitlab.lrz.de/cie_sam_public/fcmlab/ In section 3.2 we then apply the sharp interface approach on two point cloud examples taken from a benchmark-series of point clouds [31] which was built for the evaluation of reconstruction algorithms to demonstrate that the algorithm is capable of capturing also examples with convex, concave and non-smoothly boundaries involving kinks. ...
... The vertical load, prestress and thickness are given by p, σ 0 and t. The point clouds in question are taken from the benchmark by Ohrhallinger et al. [31] and may be found at https://gitlab.com/stefango74/curve-benchmark. The point clouds are scaled such that they fit into the solution domain [−1.1, 1.1] × [−1.1, 1.1] which is discretized by 16x16 elements of polynomial degree 10. ...
This paper develops and investigates a new method for the application of Dirichlet boundary conditions for computational models defined by point clouds. Point cloud models often stem from laser or structured-light scanners which are used to scan existing mechanical structures for which CAD models either do not exist or from which the artifact under investigation deviates in shape or topology. Instead of reconstructing a CAD model from point clouds via surface reconstruction and a subsequent boundary conforming mesh generation, a direct analysis without pre-processing is possible using embedded domain finite element methods. These methods use non-boundary conforming meshes which calls for a weak enforcement of Dirichlet boundary conditions. For point cloud based models, Dirichlet boundary conditions are usually imposed using a diffuse interface approach. This leads to a significant computational overhead due to the necessary computation of domain integrals. Additionally, undesired side effects on the gradients of the solution arise which can only be controlled to some extent. This paper develops a new sharp interface approach for point cloud based models which avoids both issues. The computation of domain integrals is circumvented by an implicit approximation of corresponding Voronoi diagrams of higher order and the resulting sharp approximation avoids the side-effects of diffuse approaches. Benchmark examples from the graphics as well as the computational mechanics community are used to verify the algorithm. All algorithms are implemented in the FCMLab framework and provided at https://gitlab.lrz.de/cie_sam_public/fcmlab/. Further, we discuss challenges and limitations of point cloud based analysis w.r.t. application of Dirichlet boundary conditions.
The approximation of point clouds in terms of parametric representations is a fundamental task for geometric modeling and processing applications to properly analyze the re-constructed model in its full complexity. A necessary and key step in this process requires the identification of a suitable data parameterization. If the data connectivity is determined by a rectilinear grid, standard closed-form methods are available for general multivariate configurations. Existing data-driven parameterization methods instead are limited to the univariate case and require pre/post-processing steps of the input data to handle point sequences without fixed length. In this paper we propose a dimension independent method based on Convolutional Neural Networks to assign parameter values to gridded point clouds of arbitrary size, without the need for additional data processing steps. We train the proposed networks by considering polynomial least squares approximations and demonstrate, both in the univariate and bivariate settings, that the accuracy of the final model properly scales when uniform and adaptive spline refinement is considered. A selection of numerical experiments on point clouds of different sizes highlights the performance of our parameterization schemes. Noisy data sets which simulate measurement errors are also considered.
