Dual Marching Cubes: Primal Contouring of Dual Grids

We present a method for contouring an implicit function using a grid topologically dual to structured grids such as octrees. By aligning the vertices of the dual grid with the features of the implicit function, we are able to reproduce thin features of the extracted surface without excessive subdivision required by methods such as Marching Cubes or Dual Contouring. Dual Marching Cubes produces a crack‐free, adaptive polygonalization of the surface that reproduces sharp features. Our approach maintains the advantage of using structured grids for operations such as CSG while being able to conform to the relevant features of the implicit function yielding much sparser polygonalizations than has been possible using structured grids.     

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.     

Fiber Surfaces: Generalizing Isosurfaces to Bivariate Data

Scientific visualization has many effective methods for examining and exploring scalar and vector fields, but rather fewer for bivariate fields. We report the first general purpose approach for the interactive extraction of geometric separating surfaces in bivariate fields. This method is based on fiber surfaces: surfaces constructed from sets of fibers, the multivariate analogues of isolines. We show simple methods for fiber surface definition and extraction. In particular, we show a simple and efficient fiber surface extraction algorithm based on Marching Cubes. We also show how to construct fiber surfaces interactively with geometric primitives in the range of the function. We then extend this to build user interfaces that generate parameterized families of fiber surfaces with respect to arbitrary polygons. In the special case of isovalue‐gradient plots, fiber surfaces capture features geometrically for quantitative analysis that have previously only been analysed visually and qualitatively using multi‐dimensional transfer functions in volume rendering. We also demonstrate fiber surface extraction on a variety of bivariate data.     

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).     

离散Marching Cubes算法在骨科手术模拟系统的应用

为保证模型相对准确和满足模拟手术过程中三维交互的实时性要求,手术模拟系统要求在保持模型拓扑结构的前提下简化模型。该文详细介绍了离散MarchingCubes(DiscMC)算法及其实现,在实现过程中使用查表法,解决了二义性问题,提高了程序的运行效率。实验表明,DiscMC算法在保持模型的拓扑结构基础上大幅度减少了三角面片数目,缩减比例达66%。DiscMC算法作为计算机模拟骨科手术系统的三维表面模型重构和简化算法是合适的。     

基于Double Marching Cubes的表面重建算法

为克服基于Marching Cubes的表面重建算法在绘制三维表面时因二义性面的存在而使生成的表面网格易出现错误连接而形成层间空洞的不足,提出了基于Double Marching Cubes的表面重建算法.该算法采用双立方体体素作为生成表面网格的基本单元,以双立方体的12个特征点的标记情况为依据,建立一个双立方体体素索引表,通过查找索引表的方法绘制三维表面.该算法在建立双立方体索引表时就排除掉了二义性面的所有错误连接方式,因而生成的表面网格不会出现层间空洞,避免了为消除二义性面所进行的复杂计算,加快了表面重建的速度.     

Fast Generation of Leakproof Surfaces from Well-Defined Objects by a Modified Marching Cubes Algorithm

Local surface reconstruction by the Marching Cubes algorithm and its derivatives has a well known ambiguity, which prevents constructed surfaces from being closed and simple. We investigate this ambiguity assuming that a 3D image samples well-defined objects. In this case it is justified to aim at tiling of extracted object voxels rather than at reconstructing iso surfaces. Compared to iso surface reconstruction, our algorithm provides essentially the same level of confidence with respect to surface location at a lower computational cost. We present a leak detection and mending scheme which resolves the Marching Cubes ambiguity and guarantees a well-defined behaviour with respect to which objects are covered by which surface. We detail how to implement our leak mending method within a completely tabulated Marching Cubes algorithm. We finally give an example of how the adapted algorithm is of benefit to a recently developed 3D MR spectroscopy technique.     

基于图像分割的三维重构

三维重构方法是医学图像可视化系统、治疗计划系统的重要技术。基于图像分割的三维重构方法结合了图像分割、等值面抽取、网格简化三种技术,是不同于传统Marching Cubes算法的一种三维重构方法。它首先将医学图像分割为二值图,然后利用Marching Cubes方法进行等值面抽取,最后对得到的网格模型进行简化。实验结果表明,基于图像分割的三维重构方法加快了Marching Cubes的运算速度,改善了重构的效果,有利于实现对基于三维重构的大型几何模型的实时绘制和交互。     

Controlled topology simplification

We present a simple, robust, and practical method for object simplification for applications where gradual elimination of high frequency details is desired. This is accomplished by converting an object into multi resolution volume rasters using a controlled filtering and sampling technique. A multiresolution triangle mesh hierarchy can then be generated by applying the Marching Cubes algorithm. We further propose an adaptive surface generation algorithm to reduce the number of triangles generated by the standard Marching Cubes. Our method simplifies the topology of objects in a controlled fashion. In addition, at each level of detail, multilayered meshes can be used for an efficient antialiased rendering     

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.     

Marching Cubes算法研究现状

对现有的Marching Cubes改进算法从拓扑结构二义性、提高逼近精度、算法的时间和空间效率3个方面进行综述,对每一类改进算法进行新的分类,并对各类算法的实验结果进行比较.     

Convex contouring of volumetric data

In this paper, we present a fast, table-driven isosurface extraction technique on volumetric data. Unlike Marching Cubes or other cell-based algorithms, the proposed polygonization generates convex negative space inside individual cells, enabling fast collision detection on the triangulated isosurface. In our implementation, we are able to perform over 2 million point classifications per second. The algorithm is driven by an automatically constructed lookup table that stores compact decision trees by sign configurations. The decision trees determine triangulations dynamically by values at cell corners. Using the same technique, we can perform fast, crack-free multiresolution contouring on nested grids of volumetric data. The method can also be extended to extract isosurfaces on arbitrary convex, space-filling polyhedra.     

MC算法与MT算法的比较

三维医学图像表面重建算法是近来的一个研究热点,目前已有多种算法,其中最具代表性的是Marching Cubes(MC)算法和Marching Tetrahedra(MT)算法.本文对这两种算法的实现原理进行了分析比较,给出了实现这两种算法的数据结构,分析了两者的实现过程,并针对具体的实验结果比较了两者的优劣.     

Parallel Surface Reconstruction for Particle‐Based Fluids

This paper presents a novel method that improves the efficiency of high‐quality surface reconstructions for particle‐based fluids using Marching Cubes. By constructing the scalar field only in a narrow band around the surface, the computational complexity and the memory consumption scale with the fluid surface instead of the volume. Furthermore, a parallel implementation of the method is proposed. The presented method works with various scalar field construction approaches. Experiments show that our method reconstructs high‐quality surface meshes efficiently even on single‐core CPUs. It scales nearly linearly on multi‐core CPUs and runs up to fifty times faster on GPUs compared to the original scalar field construction approaches.     

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 .     

Spherical Parameterization of Marching Cubes IsoSurfaces Based upon Nearest Neighbor Coordinates

We present some new methods for parameterizing the triangle mesh surface (TMS) which result from the Marching Cubes (MC) algorithm. The methods apply to surfaces of genus zero and the parameter domain is a unit sphere. We take advantage of some special properties of the TMS resulting from the MC algorithm to obtain simple, computational efficient representations of the nearest neighbor coordinates and utilize these coordinates in the characterization of the parameterization by means of systems of equatio...     

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.     

High‐speed Marching Cubes using HistoPyramids

We present an implementation approach for Marching Cubes (MC) on graphics hardware for OpenGL 2.0 or comparable graphics APIs. It currently outperforms all other known graphics processing units (GPU)‐based iso‐surface extraction algorithms in direct rendering for sparse or large volumes, even those using the recently introduced geometry shader (GS) capabilites. To achieve this, we outfit the Histogram Pyramid (HP) algorithm, previously only used in GPU data compaction, with the capability for arbitrary data expansion. After reformulation of MC as a data compaction and expansion process, the HP algorithm becomes the core of a highly efficient and interactive MC implementation. For graphics hardware lacking GSs, such as mobile GPUs, the concept of HP data expansion is easily generalized, opening new application domains in mobile visual computing. Further, to serve recent developments, we present how the HP can be implemented in the parallel programming language CUDA (compute unified device architecture), by using a novel 1D chunk/layer construction.     

Edge groups: an approach to understanding the mesh quality of marching methods

Marching Cubes is the most popular isosurface extraction algorithm due to its simplicity, efficiency and robustness. It has been widely studied, improved, and extended. While much early work was concerned with efficiency and correctness issues, lately there has been a push to improve the quality of Marching Cubes meshes so that they can be used in computational codes. In this work we present a new classification of MC cases that we call Edge Groups, which helps elucidate the issues that impact the triangle quality of the meshes that the method generates. This formulation allows a more systematic way to bound the triangle quality, and is general enough to extend to other polyhedral cell shapes used in other polygonization algorithms. Using this analysis, we also discuss ways to improve the quality of the resulting triangle mesh, including some that require only minor modifications of the original algorithm.