An Optimal Transport Approach to Robust Reconstruction and Simplification of 2D Shapes

We propose a robust 2D shape reconstruction and simplification algorithm which takes as input a defect‐laden point set with noise and outliers. We introduce an optimal‐transport driven approach where the input point set, considered as a sum of Dirac measures, is approximated by a simplicial complex considered as a sum of uniform measures on 0‐ and 1‐simplices. A fine‐to‐coarse scheme is devised to construct the resulting simplicial complex through greedy decimation of a Delaunay triangulation of the input point set. Our method performs well on a variety of examples ranging from line drawings to grayscale images, with or without noise, features, and boundaries.     

Feature‐Preserving Reconstruction of Singular Surfaces

Reconstructing a surface mesh from a set of discrete point samples is a fundamental problem in geometric modeling. It becomes challenging in presence of ‘singularities’ such as boundaries, sharp features, and non‐manifolds. A few of the current research in reconstruction have addressed handling some of these singularities, but a unified approach to handle them all is missing. In this paper we allow the presence of various singularities by requiring that the sampled object is a collection of smooth surface patches with boundaries that can meet or intersect. Our algorithm first identifies and reconstructs the features where singularities occur. Next, it reconstructs the surface patches containing these feature curves. The identification and reconstruction of feature curves are achieved by a novel combination of the Gaussian weighted graph Laplacian and the Reeb graphs. The global reconstruction is achieved by a method akin to the well known Cocone reconstruction, but with weighted Delaunay triangulation that allows protecting the feature samples with balls. We provide various experimental results to demonstrate the effectiveness of our feature‐preserving singular surface reconstruction algorithm.     

Feature‐Preserving Surface Reconstruction From Unoriented,Noisy Point Data

We propose a robust method for surface mesh reconstruction from unorganized, unoriented, noisy and outlier‐ridden 3D point data. A kernel‐based scale estimator is introduced to estimate the scale of inliers of the input data. The best tangent planes are computed for all points based on mean shift clustering and adaptive scale sample consensus, followed by detecting and removing outliers. Subsequently, we estimate the normals for the remaining points and smooth the noise using a surface fitting and projection strategy. As a result, the outliers and noise are removed and filtered, while the original sharp features are well preserved. We then adopt an existing method to reconstruct surface meshes from the processed point data. To preserve sharp features of the generated meshes that are often blurred during reconstruction, we describe a two‐step approach to effectively recover original sharp features. A number of examples are presented to demonstrate the effectiveness and robustness of our method.     

Multi-scale tensor voting for feature extraction from unstructured point clouds

Identifying sharp features in a 3D model is essential for shape analysis, matching and a wide range of geometry processing applications. This paper presents a new method based on the tensor voting theory to extract sharp features from an unstructured point cloud which may contain random noise, outliers and artifacts. Our method first takes the voting tensors at every point using the corresponding neighborhoods and computes the feature weight to infer the local structure via eigenvalue analysis of the tensor. The optimal scale for a point is automatically determined by observing the feature weight variation in order to deal with both a noisy smooth region and a sharp edge. We finally extract the points at sharp features using adaptive thresholding of the feature weight and the feature completion process. The multi-scale tensor voting of a given point set improves noise sensitivity and scale dependency of an input model. We demonstrate the strength of the proposed method in terms of efficiency and robustness by comparing it with other feature detection algorithms.     

Feature Preserving Mesh Generation from 3D Point Clouds

We address the problem of generating quality surface triangle meshes from 3D point clouds sampled on piecewise smooth surfaces. Using a feature detection process based on the covariance matrices of Voronoi cells, we first extract from the point cloud a set of sharp features. Our algorithm also runs on the input point cloud a reconstruction process, such as Poisson reconstruction, providing an implicit surface. A feature preserving variant of a Delaunay refinement process is then used to generate a mesh approximating the implicit surface and containing a faithful representation of the extracted sharp edges. Such a mesh provides an enhanced trade‐off between accuracy and mesh complexity. The whole process is robust to noise and made versatile through a small set of parameters which govern the mesh sizing, approximation error and shape of the elements. We demonstrate the effectiveness of our method on a variety of models including laser scanned datasets ranging from indoor to outdoor scenes.     

Orienting raw point sets by global contraction and visibility voting

We present a global method for consistently orienting a defective raw point set with noise, non-uniformities and thin sharp features. Our method seamlessly combines two simple but effective techniques—constrained Laplacian smoothing and visibility voting—to tackle this challenge. First, we apply a Laplacian contraction to the given point cloud, which shrinks the shape a little bit. Each shrunk point corresponds to an input point and shares a visibility confidence assigned by voting from multiple viewpoints. The confidence is increased (resp. decreased) if the input point (resp. its corresponding shrunk point) is visible. Then, the initial normals estimated by principal component analysis are flipped according to the contraction vectors from shrunk points to the corresponding input points and the visibility confidence. Finally, we apply a Laplacian smoothing twice to correct the orientation of points with zero or low confidence. Our method is conceptually simple and easy to implement, without resorting to any complicated data structures and advanced solvers. Numerous experiments demonstrate that our method can orient the defective raw point clouds in a consistent manner. By taking advantage of our orientation information, the classical implicit surface reconstruction algorithms can faithfully generate the surface.     

Sharp feature preserving MLS surface reconstruction based on local feature line approximations

Sharp features in manufactured and designed objects require particular attention when reconstructing surfaces from unorganized scan point sets using moving least squares (MLS) fitting. It is an inherent property of MLS fitting that sharp features are smoothed out. Instead of searching for appropriate new fitting functions our approach computes a modified local point neighborhood so that a standard MLS fitting can be applied enhanced by sharp features reconstruction.We present a two-stage algorithm. In a pre-processing step sharp feature points are marked first. This algorithm is robust to noise since it is based on Gauss map clustering. In the main phase, the selected feature points are used to locally approximate the feature curve and to segment and enhance the local point neighborhood. The MLS projection thus leads to a piecewise smooth surface preserving all sharp features. The method is simple to implement and able to preserve line-type features as well as corner-type features during reconstruction.     

基于曲面局平特性的散乱数据拓扑重建算法

提出了一种基于曲面局平特性的,以散乱点集及其密度指标作为输入,以三角形分片线性曲面作为输出的拓扑重建算法.算法利用曲面的局平特性,从散乱点集三维Delaunay三角剖分的邻域结构中完成每个样点周围的局部拓扑重建,并从局部重建的并集中删除不相容的三角形,最终得到一个二维流形拓扑曲面集作为重建结果.该算法适应于包括单侧曲面在内的任意不自交的拓扑曲面集,并且重建结果是相对优化的曲面三角形剖分,可以应用于科学计算可视化、雕塑曲面造型和反求工程等领域.     

Robust and Efficient Implicit Surface Reconstruction for Point Clouds Based on Convexified Image Segmentation

We present an implicit surface reconstruction algorithm for point clouds. We view the implicit surface reconstruction as a three dimensional binary image segmentation problem that segments the entire space $\mathbb R ^3$ or the computational domain into an interior region and an exterior region while the boundary between these two regions fits the data points properly. The key points with using an image segmentation formulation are: (1) an edge indicator function that gives a sharp indicator of the surface location, and (2) an initial image function that provides a good initial guess of the interior and exterior regions. In this work we propose novel ways to build both functions directly from the point cloud data. We then adopt recent convexified image segmentation models and fast computational algorithms to achieve efficient and robust implicit surface reconstruction for point clouds. We test our methods on various data sets that are noisy, non-uniform, and with holes or with open boundaries. Moreover, comparisons are also made to current state of the art point cloud surface reconstruction techniques.     

Extracting lines of curvature from noisy point clouds

We present a robust framework for extracting lines of curvature from point clouds. First, we show a novel approach to denoising the input point cloud using robust statistical estimates of surface normal and curvature which automatically rejects outliers and corrects points by energy minimization. Then the lines of curvature are constructed on the point cloud with controllable density. Our approach is applicable to surfaces of arbitrary genus, with or without boundaries, and is statistically robust to noise and outliers while preserving sharp surface features. We show our approach to be effective over a range of synthetic and real-world input datasets with varying amounts of noise and outliers. The extraction of curvature information can benefit many applications in CAD, computer vision and graphics for point cloud shape analysis, recognition and segmentation. Here, we show the possibility of using the lines of curvature for feature-preserving mesh construction directly from noisy point clouds.     

Implicit and Nonparametric Shape Reconstruction from Unorganized Data Using a Variational Level Set Method

In this paper we consider a fundamental visualization problem: shape reconstruction from an unorganized data set. A new minimal-surface-like model and its variational and partial differential equation (PDE) formulation are introduced. In our formulation only distance to the data set is used as our input. Moreover, the distance is computed with optimal speed using a new numerical PDE algorithm. The data set can include points, curves, and surface patches. Our model has a natural scaling in the nonlinear regularization that allows flexibility close to the data set while it also minimizes oscillations between data points. To find the final shape, we continuously deform an initial surface following the gradient flow of our energy functional. An offset (an exterior contour) of the distance function to the data set is used as our initial surface. We have developed a new and efficient algorithm to find this initial surface. We use the level set method in our numerical computation in order to capture the deformation of the initial surface and to find an implicit representation (using the signed distance function) of the final shape on a fixed rectangular grid. Our variational/PDE approach using the level set method allows us to handle complicated topologies and noisy or highly nonuniform data sets quite easily. The constructed shape is smoother than any piecewise linear reconstruction. Moreover, our approach is easily scalable for different resolutions and works in any number of space dimensions.     

Robust normal estimation and region growing segmentation of infrastructure 3D point cloud models

Modern remote sensing technologies such as three-dimensional (3D) laser scanners and image-based 3D scene reconstruction are in increasing demand for applications in civil infrastructure design, maintenance, operation, and as-built construction verification. The complex nature of the 3D point clouds these technologies generate, as well as the often massive scale of the 3D data, make it inefficient and time consuming to manually analyze and manipulate point clouds, and highlights the need for automated analysis techniques. This paper presents one such technique, a new region growing algorithm for the automated segmentation of both planar and non-planar surfaces in point clouds. A core component of the algorithm is a new point normal estimation method, an essential task for many point cloud processing algorithms. The newly developed estimation method utilizes robust multivariate statistical outlier analysis for reliable normal estimation in complex 3D models, considering that these models often contain regions of varying surface roughness, a mixture of high curvature and low curvature regions, and sharp features. An adaptation of Mahalanobis distance, in which the mean vector and covariance matrix are derived from a high-breakdown multivariate location and scale estimator called Deterministic MM-estimator (DetMM) is used to find and discard outlier points prior to estimating the best local tangent plane around any point in a cloud. This approach is capable of more accurately estimating point normals located in highly curved regions or near sharp features. Thereafter, the estimated point normals serve a region growing segmentation algorithm that only requires a single input parameter, an improvement over existing methods which typically require two control parameters. The reliability and robustness of the normal estimation subroutine was compared against well-known normal estimation methods including the Minimum Volume Ellipsoid (MVE) and Minimum Covariance Determinant (MCD) estimators, along with Maximum Likelihood Sample Consensus (MLESAC). The overall region growing segmentation algorithm was then experimentally validated on several challenging 3D point clouds of real-world infrastructure systems. The results indicate that the developed approach performs more accurately and robustly in comparison with conventional region growing methods, particularly in the presence of sharp features, outliers and noise.     

Robust reconstruction of 2D curves from scattered noisy point data

In this paper, a robust algorithm is proposed for reconstructing 2D curve from unorganized point data with a high level of noise and outliers. By constructing the quadtree of the input point data, we extract the “grid-like” boundaries of the quadtree, and smooth the boundaries using a modified Laplacian method. The skeleton of the smoothed boundaries is computed and thereby the initial curve is generated by circular neighboring projection. Subsequently, a normal-based processing method is applied to the initial curve to smooth jagged features at low curvatures areas, and recover sharp features at high curvature areas. As a result, the curve is reconstructed accurately with small details and sharp features well preserved. A variety of experimental results demonstrate the effectiveness and robustness of our method.     

Spline-based feature curves from point-sampled geometry

Defining sharp features in a 3D model facilitates a better understanding of the surface and aids geometric processing and graphics applications, such as reconstruction, filtering, simplification, reverse engineering, visualization, and non-photo realism. We present a robust method that identifies sharp features in a point-based model by returning a set of smooth spline curves aligned along the edges. Our feature extraction leverages the concepts of robust moving least squares to locally project points to potential features. The algorithm processes these points to construct arc-length parameterized spline curves fit using an iterative refinement method, aligning smooth and continuous curves through the feature points. We demonstrate the benefits of our method with three applications: surface segmentation, surface meshing and point-based compression.     

规则特征曲面点云法向估计

目的 针对特征曲面点云法矢估计不准确,点云处理时容易丢失曲面的细节特征等问题,提出基于高斯映射的特征曲面散乱点云法向估计法。方法 首先,用主成分分析法粗略地估算点云法向和特征点;其次,将特征点的各向同性邻域映射到高斯球,用K均值聚类法对高斯球上的数据分割成多个子集,以最优子集对应的各向异性邻域拟合曲面来精确估算特征点的法向量;最后,通过测试估计法向与标准法向的误差来评价估计法矢的准确性,并且将估计的法向应用到点云曲面重建中来比较特征保留效果。结果 本文方法估计的法向最小误差接近0,对噪声有较好的鲁棒性,重建的曲面能保留曲面的尖锐特征,相比于其他法向估计法,所提出的方法估计的法向更准确。结论 本文方法能够比较准确的估算尖锐特征曲面法向量,对噪声鲁棒性强,具有较高的适用性。     

Fitting Sharp Features with Loop Subdivision Surfaces

Various methods have been proposed for fitting subdivision surfaces to different forms of shape data (e.g., dense meshes or point clouds), but none of these methods effectively deals with shapes with sharp features, that is, creases, darts and corners. We present an effective method for fitting a Loop subdivision surface to a dense triangle mesh with sharp features. Our contribution is a new exact evaluation scheme for the Loop subdivision with all types of sharp features, which enables us to compute a fitting Loop subdivision surface for shapes with sharp features in an optimization framework. With an initial control mesh obtained from simplifying the input dense mesh using QEM, our fitting algorithm employs an iterative method to solve a nonlinear least squares problem based on the squared distances from the input mesh vertices to the fitting subdivision surface. This optimization framework depends critically on the ability to express these distances as quadratic functions of control mesh vertices using our exact evaluation scheme near sharp features. Experimental results are presented to demonstrate the effectiveness of the method.     

Point set silhouettes via local reconstruction

We present an algorithm to compute the silhouette set of a point cloud. Previous methods extract point set silhouettes by thresholding point normals, which can lead to simultaneous over- and under-detection of silhouettes. We argue that additional information such as surface curvature is necessary to resolve these issues. To this end, we develop a local reconstruction scheme using Gabriel and intrinsic Delaunay criteria and define point set silhouettes based on the notion of a silhouette-generating set. The mesh umbrellas, or local reconstructions of one-ring triangles surrounding each point sample, generated by our method enable accurate silhouette identification near sharp features and close-by surface sheets, and provide the information necessary to detect other characteristic curves such as creases and boundaries. We show that these curves collectively provide a sparse and intuitive visualisation of point-cloud data.     

Bipartite Polar Classification for Surface Reconstruction

In this paper, we propose bipartite polar classification to augment an input unorganized point set ? with two disjoint groups of points distributed around the ambient space of ? to assist the task of surface reconstruction. The goal of bipartite polar classification is to obtain a space partitioning of ? by assigning pairs of Voronoi poles into two mutually invisible sets lying in the opposite sides of ? through direct point set visibility examination. Based on the observation that a pair of Voronoi poles are mutually invisible, spatial classification is accomplished by carving away visible exterior poles with their counterparts simultaneously determined as interior ones. By examining the conflicts of mutual invisibility, holes or boundaries can also be effectively detected, resulting in a hole‐aware space carving technique. With the classified poles, the task of surface reconstruction can be facilitated by more robust surface normal estimation with global consistent orientation and off‐surface point specification for variational implicit surface reconstruction. We demonstrate the ability of the bipartite polar classification to achieve robust and efficient space carving on unorganized point clouds with holes and complex topology and show its application to surface reconstruction.     

Similarity-guided streamline placement with error evaluation

Most streamline generation algorithms either provide a particular density of streamlines across the domain or explicitly detect features, such as critical points, and follow customized rules to emphasize those features. However, the former generally includes many redundant streamlines, and the latter requires Boolean decisions on which points are features (and may thus suffer from robustness problems for real-world data). We take a new approach to adaptive streamline placement for steady vector fields in 2D and 3D. We define a metric for local similarity among streamlines and use this metric to grow streamlines from a dense set of candidate seed points. The metric considers not only Euclidean distance, but also a simple statistical measure of shape and directional similarity. Without explicit feature detection, our method produces streamlines that naturally accentuate regions of geometric interest. In conjunction with this method, we also propose a quantitative error metric for evaluating a streamline representation based on how well it preserves the information from the original vector field. This error metric reconstructs a vector field from points on the streamline representation and computes a difference of the reconstruction from the original vector field.