Contouring Discrete Indicator Functions

We present a method for calculating the boundary of objects from Discrete Indicator Functions that store 2‐material volume fractions with a high degree of accuracy. Although Marching Cubes and its derivatives are effective methods for calculating contours of functions sampled over discrete grids, these methods perform poorly when contouring non‐smooth functions such as Discrete Indicator Functions. In particular, Marching Cubes will generate surfaces that exhibit aliasing and oscillations around the exact surface. We derive a simple solution to remove these problems by using a new function to calculate the positions of vertices along cell edges that is efficient, easy to implement, and does not require any optimization or iteration. Finally, we provide empirical evidence that the error introduced by our contouring method is significantly less than is introduced by Marching Cubes.     

Efficient and Quality Contouring Algorithms on the GPU

Interactive isosurface extraction has recently become possible through successful efforts to map algorithms such as Marching Cubes (MC) and Marching Tetrahedra (MT) to modern Graphics Processing Unit (GPU) architectures. Other isosurfacing algorithms, however, are not so easily portable to GPUs, either because they involve more complex operations or because they are not based on discrete case tables, as is the case with most marching techniques. In this paper, we revisit the Dual Contouring (MC) and Macet isosurface extraction algorithms and propose, respectively: (i) a novel, efficient and parallelizable version of Dual Contouring and (ii) a set of GPU modules which extend the original Marching Cubes algorithm. Similar to marching methods, our novel technique is based on a case table, which allows for a very efficient GPU implementation. In addition, we enumerate and evaluate several alternatives to implement efficient contouring algorithms on the GPU, and present trade‐offs among all approaches. Finally, we validate the efficiency and quality of the tessellations produced in all these alternatives.     

Edge Transformations for Improving Mesh Quality of Marching Cubes

Marching Cubes is a popular choice for isosurface extraction from regular grids due to its simplicity, robustness, and efficiency. One of the key shortcomings of this approach is the quality of the resulting meshes, which tend to have many poorly shaped and degenerate triangles. This issue is often addressed through post processing operations such as smoothing. As we demonstrate in experiments with several datasets, while these improve the mesh, they do not remove all degeneracies, and incur an increased and unbounded error between the resulting mesh and the original isosurface. Rather than modifying the resulting mesh, we propose a method to modify the grid on which Marching Cubes operates. This modification greatly increases the quality of the extracted mesh. In our experiments, our method did not create a single degenerate triangle, unlike any other method we experimented with. Our method incurs minimal computational overhead, requiring at most twice the execution time of the original Marching Cubes algorithm in our experiments. Most importantly, it can be readily integrated in existing Marching Cubes implementations, and is orthogonal to many Marching Cubes enhancements (particularly, performance enhancements such as out-of-core and acceleration structures).     

The Mesh Propagation Algorithm for Isosurface Construction

A new algorithm, Mesh Propagation, is presented for the generation of isosurfaces from three-dimensional discrete data sets. While producing the same surface mesh as that generated by a corrected Marching Cubes algorithm, its characteristic is that it constructs an isosurface using connected strips of dynamically triangulated polygons. This compact data structure speeds up surface construction and reduces surface storage requirements. The surface can also be displayed more quickly, particularly where there is hardware support for rendering triangle strips. With engineering as well as medical imaging applications in mind, the algorithm can be used with both irregular and rectilinear grids of data, the primitive volume elements need not be hexahedral only, and volumes of heterogeneous polyhedral elements are supported without traversal complications. The algorithm propagates through the cells in the grid and uses the same lookup table topologies as Marching Cubes to determine patches of surface-element intersection; additional tables are used for non-hexahedral elements. The surface patches are dynamically coded into triangle strips which are then concatenated and linked to construct the surface. The data structures used for propagating through the volume overcome the topological ambiguities associated with table-based methods of surface construction and no holes are generated in the final mesh.     

A reverse engineering system for rapid manufacturing of complex objects

This paper presents a reverse engineering system for rapid modeling and manufacturing of products with complex surfaces. The system consists of three main components: a 3D optical digitizing system, a surface reconstruction software and a rapid prototyping machine. The unique features of the 3D optical digitizing system include the use of white-light source, and cost-effective and quick image acquisition. The surface reconstruction process consists of three major steps: (1) range view registration by an iterative closed-form solution, (2) range surface integration by reconstructing an implicit function to update the volumetric grid, and (3) iso-surface extraction by the Marching Cubes algorithm. The modeling software exports models in STL format, which are used as input to an FDM 2000 machine to manufacture products. The examples are included to illustrate the systems and the methods.     

SSD: Smooth Signed Distance Surface Reconstruction

We introduce a new variational formulation for the problem of reconstructing a watertight surface defined by an implicit equation, from a finite set of oriented points; a problem which has attracted a lot of attention for more than two decades. As in the Poisson Surface Reconstruction approach, discretizations of the continuous formulation reduce to the solution of sparse linear systems of equations. But rather than forcing the implicit function to approximate the indicator function of the volume bounded by the implicit surface, in our formulation the implicit function is forced to be a smooth approximation of the signed distance function to the surface. Since an indicator function is discontinuous, its gradient does not exist exactly where it needs to be compared with the normal vector data. The smooth signed distance has approximate unit slope in the neighborhood of the data points. As a result, the normal vector data can be incorporated directly into the energy function without implicit function smoothing. In addition, rather than first extending the oriented points to a vector field within the bounding volume, and then approximating the vector field by a gradient field in the least squares sense, here the vector field is constrained to be the gradient of the implicit function, and a single variational problem is solved directly in one step. The formulation allows for a number of different efficient discretizations, reduces to a finite least squares problem for all linearly parameterized families of functions, and does not require boundary conditions. The resulting algorithms are significantly simpler and easier to implement, and produce results of quality comparable with state‐of‐the‐art algorithms. An efficient implementation based on a primal‐graph octree‐based hybrid finite element‐finite difference discretization, and the Dual Marching Cubes isosurface extraction algorithm, is shown to produce high quality crack‐free adaptive manifold polygon meshes.     

Dual Sheet Meshing: An Interactive Approach to Robust Hexahedralization

The combinatorial dual of a hex mesh induces a collection of mutually intersecting surfaces (dual sheets). Inspired by Campen et al.'s work on quad meshing [CBK12, CK14], we propose to directly generate such dual sheets so that, as long as the volume is properly partitioned by the dual sheets, we are guaranteed to arrive at a valid all‐hex mesh topology. Since automatically generating dual sheets seems much harder than the 2D counterpart, we chose to leave the task to the user; our system is equipped with a few simple 3D modeling tools for interactively designing dual sheets. Dual sheets are represented as implicit surfaces in our approach, greatly simplifying many of the computational steps such as finding intersections and analyzing topology. We also propose a simple algorithm for primalizing the dual graph where each dual cell, often enclosing singular edges, gets mapped onto a reference polyhedron via harmonic parameterization. Preservation of sharp features is simply achieved by modifying the boundary conditions. We demonstrate the feasibility of our approach through various modeling examples.     

一种面向三维点集的快速表面重构算法

在对目前比较流行的空间三角化算法进行对比研究的基础上 ,对 Hugues Hoppe提出的算法进行了改进 ,即借鉴 Marching Cubes算法的基本思想 ,首先通过自动选取适当的参数 ,用包围盒方法将三维散乱点划分为数据区域 ;然后求取点的切平面及法向 ,同时采用广度优先算法遍历数据点来调整法向和快速地求取 Marching Cubes的等势函数 ;最后用基于查表法的 Marching Cubes来输出三角面片 ,即得到表面模型 .实验结果表明 ,改进后的算法效率有较大的提高 .新算法不仅适用于表面三维散乱点数据 ,也可以对体数据进行重构 ,具有一定的通用性 .     

Mesh composition on models with arbitrary boundary topology

This paper presents a new approach for the mesh composition on models with arbitrary boundary topology. After cutting the needed parts from existing mesh models and putting them into the right pose, an implicit surface is adopted to smoothly interpolate the boundaries of models under composition. An interface is developed to control the shape of the implicit transient surface by using sketches to specify the expected silhouettes. After that, a localized Marching Cubes algorithm is investigated to tessellate the implicit transient surface so that the mesh surface of composed model is generated. Different from existing approaches in which the models under composition are required to have pairwise merging boundaries, the framework developed based on our techniques have the new function to fuse models with arbitrary boundary topology.     

Newton-Krylov-FAC methods for problems discretized on locally refined grids

Many problems in computational science and engineering are nonlinear and time-dependent. The solutions to these problems may include spatially localized features, such as boundary layers or sharp fronts, that require very fine grids to resolve. In many cases, it is impractical or prohibitively expensive to resolve these features with a globally fine grid, especially in three dimensions. Adaptive mesh refinement (AMR) is a dynamic gridding approach that employs a fine grid only where necessary to resolve such features. Numerous AMR codes exist for solving hyperbolic problems with explicit time stepping and some classes of linear elliptic problems. Researchers have paid much less attention to the development of AMR algorithms for the implicit solution of systems of nonlinear equations. Recent efforts encompassing a variety of applications demonstrate that Newton-Krylov methods are effective when combined with multigrid preconditioners. This suggests that hierarchical methods, such as the Fast Adaptive Composite grid (FAC) method of McCormick and Thomas, can provide effective preconditioning for problems discretized on locally refined grids. In this paper, we address algorithm and implementation issues for the use of Newton-Krylov-FAC methods on structured AMR grids. In our software infrastructure, we combine nonlinear solvers from KINSOL and PETSc with the SAMRAI AMR library, and include capabilities for implicit time stepping. We have obtained convergence rates independent of the number of grid refinement levels for simple, nonlinear, Poisson-like, problems. Additional efforts to employ this infrastructure in new applications are underway. Communicated by: G. Wittum     

反求工程中的隐式曲面快速自适应性多边形算法

论文给出一种反求工程中基于三角形细分的隐式曲面快速自适应性多边形化方法。该文先由输入的三维扫描数据点利用空间延展的MarchingCubes方法得到隐式曲面较为粗糙的三角形表面网格形状,再利用该文的自适应性优化方法对粗糙网格从三个方面自适应性调整,即调整网格顶点法向,控制曲率,再补偿网格抽样率。从而生成的三角网格和采样点具有局部适应性,能随着曲率的变化自动控制采样点的疏密程度,消除了逼近网格中的T-形边。实验表明,恢复的隐式曲面能很好地反映形状特征,能满足反求工程的实时需求。     

Quality Isosurface Mesh Generation Using an Extended Marching Cubes Lookup Table

The Marching Cubes Algorithm may return degenerate, zero area isosurface triangles, and often returns isosurface triangles with small areas, edges or angles. We show how to avoid both problems using an extended Marching Cubes lookup table. As opposed to the conventional Marching Cubes lookup table, the extended lookup table differentiates scalar values equal to the isovalue from scalar values greater than the isovalue. The lookup table has 3⁸= 6561 entries, based on three possible labels, ‘?’ or ‘=’ or ‘+’, of each cube vertex. We present an algorithm based on this lookup table which returns an isosurface close to the Marching Cubes isosurface, but without any degenerate triangles or any small areas, edges or angles.     

隐式曲面的快速适应性多边形化算法

通过将隐式曲面多边形化过程分为“构造”和“适应性采样”两个阶段，实现了隐式曲面多边形逼近网格的适应性构造．通过基于空间延展的Marching Cubes方法得到隐式曲面较为粗糙的均匀多边形化逼近，根据曲面上的局部曲率分布，运用适应性细分规则对粗糙网格进行细分迭代，并利用梯度下降法将细分出的新顶点定位到隐式曲面上；最终得到的多边形网格是适应性的单纯复形网格，其在保持规定逼近精度的前提下，减少了冗余三角形的产生，网格质量有明显改善．该算法可用于隐式曲面的交互式可视化过程．     

Mesh filtering algorithm using an adaptive 3D convolution kernel applied to a volume-based vector distance field

This paper addresses the problem of feature preserving mesh filtering, which occurs in surface reconstruction of scanned objects, which include acquisition noise to be removed without altering sharp edges. We propose a method based on a vector field distance transform of the mesh to process. It is a volume-based implicit surface modeling, which provides an alternative representation of meshes. We use an adaptive 3D convolution kernel applied to the voxels of the distance transform model. Weights of the kernel elements are determined according to the angle between the vectors of the implicit field. We also propose a new adaptation of the Marching Cubes algorithm in order to extract the isosurface from the vector implicit field after the filtering process. We compare our method to the previous one introduced using the vector field representation and to other feature preserving adaptive filtering algorithms. According to error metric evaluations, we show that our new design provides high quality filtering results while better preserving geometric features.     

保持尖锐特征的隐式曲线绘制算法

隐式曲线在生物、医学、气象、地学、石油勘探及物探等领域有着广泛的应用。 提出一种绘制带有尖锐特征的平面隐式曲线的算法,能有效地提取隐式曲线的尖锐特征。该算 法首先确定曲线的绘制区域,采用自上而下的方式生成绘制区域的四叉树表示,并在四叉树节 点表示的每个单元格内生成一个数值场特征点;然后连接特征点生成对偶网格;最后,利用 Marching Squares 算法生成曲线。实验结果表明,该算法能在网格较稀松的情况下绘制出隐式 曲线,并且可以实现曲线的尖锐特征。     

An Algorithm for Projecting Points onto a Patched CAD Model

We are interested in building structured overlap-ping grids for geometries defined by Computer-Aided-Design (CAD) packages. Geometric information defining the boundary surfaces of a computation domain is often provided in the form of a collection of possibly hundreds of trimmed patches. The first step in building an overlapping volume grid on such a geometry is to build overlapping surface grids. A surface grid is typically built using hyperbolic grid generation; starting from a curve on the surface, a grid is grown by marching over the surface. A given hyperbolic grid will typically cover many of the underlying CAD surface patches. The fundamental operation needed for building surface grids is that of projecting a point in space onto the closest point on the CAD surface. We describe a fast and robust algorithm for performing this projection which makes use of a fairly coarse global triangulation of the CAD geometry. Before the global triangulation is constructed the connectivity of the model is determined by an edge-matching algorithm which corrects for gaps and overlaps between neighbouring patches. ID="A1" Correspondence and offprint requests to: Dr. W. D. Henshaw, Center for Applied Scientific Computing, L-661, Lawrence Livermore National Laboratory, Livermore, CA 94551, USA. E-mail: henshaw@llnl.gov     

A unifying grid approach for solving potential flows applicable to structured and unstructured grid configurations

In this study, an efficient numerical method is proposed for unifying the structured and unstructured grid approaches for solving the potential flows. The new method, named as the “alternating cell directions implicit - ACDI”, solves for the structured and unstructured grid configurations equally well. The new method in effect applies a line implicit method similar to the Line Gauss Seidel scheme for complex unstructured grids including mixed type quadrilateral and triangle cells. To this end, designated alternating directions are taken along chains of contiguous cells, i.e. ‘cell directions’, and an ADI-like sweeping is made to update these cells using a Line Gauss Seidel like scheme. The algorithm makes sure that the entire flow field is updated by traversing each cell twice at each time step for unstructured quadrilateral grids that may contain triangular cells. In this study, a cell-centered finite volume formulation of the ACDI method is demonstrated. The solutions are obtained for incompressible potential flows around a circular cylinder and a forward step. The results are compared with the analytical solutions and numerical solutions using the implicit ADI and the explicit Runge-Kutta methods on single-and multi-block structured and unstructured grids. The results demonstrate that the present ACDI method is unconditionally stable, easy to use and has the same computational performance in terms of convergence, accuracy and run times for both the structured and unstructured grids.     

High-quality extraction of isosurfaces from regular and irregular grids

Isosurfaces are ubiquitous in many fields, including visualization, graphics, and vision. They are often the main computational component of important processing pipelines (e.g. , surface reconstruction), and are heavily used in practice. The classical approach to compute isosurfaces is to apply the Marching Cubes algorithm, which although robust and simple to implement, generates surfaces that require additional processing steps to improve triangle quality and mesh size. An important issue is that in some cases, the surfaces generated by Marching Cubes are irreparably damaged, and important details are lost which can not be recovered by subsequent processing. The main motivation of this work is to develop a technique capable of constructing high-quality and high-fidelity isosurfaces. We propose a new advancing front technique that is capable of creating high-quality isosurfaces from regular and irregular volumetric datasets. Our work extends the guidance field framework of Schreiner et al. to implicit surfaces, and improves it in significant ways. In particular, we describe a set of sampling conditions that guarantee that surface features will be captured by the algorithm. We also describe an efficient technique to compute a minimal guidance field, which greatly improves performance. Our experimental results show that our technique can generate high-quality meshes from complex datasets.