A Finite Element Method on Convex Polyhedra

We present a method for animating deformable objects using a novel finite element discretization on convex polyhedra. Our finite element approach draws upon recently introduced 3D mean value coordinates to define smooth interpolants within the elements. The mathematical properties of our basis functions guarantee convergence. Our method is a natural extension to linear interpolants on tetrahedra: for tetrahedral elements, the methods are identical. For fast and robust computations, we use an elasticity model based on Cauchy strain and stiffness warping. This more flexible discretization is particularly useful for simulations that involve topological changes, such as cutting or fracture. Since splitting convex elements along a plane produces convex elements, remeshing or subdivision schemes used in simulations based on tetrahedra are not necessary, leading to less elements after such operations. We propose various operators for cutting the polyhedral discretization. Our method can handle arbitrary cut trajectories, and there is no limit on how often elements can be split.     

一种保持纹理的网面简化算法

在计算机视觉领域,三维网面的简化不仅要求保持物体形状和拓扑关系,还要求保持物体表面法线,纹理,颜色和边缘等物体特征,以使计算机视觉系统能有效地表示,描述,识别和理解物体和场景,为此讨论了一种基于边操作（边收缩,边分裂）,并具有颜色或灰度纹理特征保持的三维网面的简化算法,该算法将网面不对称最大距离作为形状改变测度,将邻域内颜色或灰度最大改变量作为纹理改变测试,从而在大量简化模型数据的同时,有效地保持了模型的几何形状,拓扑关系,颜色或灰度特征,以及网面顶点均匀分布。     

Polycube Shape Space

There are many methods proposed for generating polycube polyhedrons, but it lacks the study about the possibility of generating polycube polyhedrons. In this paper, we prove a theorem for characterizing the necessary condition for the skeleton graph of a polycube polyhedron, by which Steinitz's theorem for convex polyhedra and Eppstein's theorem for simple orthogonal polyhedra are generalized to polycube polyhedra of any genus and with non‐simply connected faces. Based on our theorem, we present a faster linear algorithm to determine the dimensions of the polycube shape space for a valid graph, for all its possible polycube polyhedrons. We also propose a quadratic optimization method to generate embedding polycube polyhedrons with interactive assistance. Finally, we provide a graph‐based framework for polycube mesh generation, quadrangulation, and all‐hex meshing to demonstrate the utility and applicability of our approach.     

Reconstructing Animated Meshes from Time‐Varying Point Clouds

In this paper, we describe a novel approach for the reconstruction of animated meshes from a series of time‐deforming point clouds. Given a set of unordered point clouds that have been captured by a fast 3‐D scanner, our algorithm is able to compute coherent meshes which approximate the input data at arbitrary time instances. Our method is based on the computation of an implicit function in ?⁴ that approximates the time‐space surface of the time‐varying point cloud. We then use the four‐dimensional implicit function to reconstruct a polygonal model for the first time‐step. By sliding this template mesh along the time‐space surface in an as‐rigid‐as‐possible manner, we obtain reconstructions for further time‐steps which have the same connectivity as the previously extracted mesh while recovering rigid motion exactly. The resulting animated meshes allow accurate motion tracking of arbitrary points and are well suited for animation compression. We demonstrate the qualities of the proposed method by applying it to several data sets acquired by real‐time 3‐D scanners.     

Exact and efficient construction of Minkowski sums of convex polyhedra with applications

We present an exact implementation of an efficient algorithm that computes Minkowski sums of convex polyhedra in R³. Our implementation is complete in the sense that it does not assume general position. Namely, it can handle degenerate input, and it produces exact results. We also present applications of the Minkowski-sum computation to answer collision and proximity queries about the relative placement of two convex polyhedra in R³. The algorithms use a dual representation of convex polyhedra, and their implementation is mainly based on the Arrangement package of Cgal, the Computational Geometry Algorithm Library. We compare our Minkowski-sum construction with the only three other methods that produce exact results we are aware of. One is a simple approach that computes the convex hull of the pairwise sums of vertices of two convex polyhedra. The second is based on Nef polyhedra embedded on the sphere, and the third is an output-sensitive approach based on linear programming. Our method is significantly faster. The results of experimentation with a broad family of convex polyhedra are reported. The relevant programs, source code, data sets, and documentation are available at http://www.cs.tau.ac.il/~efif/CD and a short movie [Fogel E, Halperin D. Video: Exact Minkowski sums of convex polyhedra. In: Proceedings of 21st annual ACM symposium on computational geometry. 2005. p. 382-3] that describes some of the concepts portrayed in this paper can be downloaded from http://www.cs.tau.ac.il/~efif/CD/Mink3d.avi.     

基于启发式搜索分离向量的凸多面体碰撞检测

碰撞检测是计算机模拟物理过程的基础，在计算机图形学、CAD／CAM、虚拟现实和机器人等领域有着广泛的应用．该文给出了一个新的用于凸多面体碰撞检测的算法——HP-jump．HP-jump建立了一个有效的碰撞检测模型用于报告物体的碰撞，同时提供了一个快速的启发式的策略用于搜索两个凸多面体的分离向量．该算法是利用凸多面体的层次表示来搜索支撑顶点对，用平衡二叉树来记录球面凸多边形的顶点，同时还利用了时间、空间相关性，这些都加速了算法的执行．该文的最后给出了HP-jump与GJK，I-COLLIDE算法的比较．     

Compressed Manifold Modes for Mesh Processing

This paper introduces compressed eigenfunctions of the Laplace‐Beltrami operator on 3D manifold surfaces. They constitute a novel functional basis, called the compressed manifold basis, where each function has local support. We derive an algorithm, based on the alternating direction method of multipliers (ADMM), to compute this basis on a given triangulated mesh. We show that compressed manifold modes identify key shape features, yielding an intuitive understanding of the basis for a human observer, where a shape can be processed as a collection of parts. We evaluate compressed manifold modes for potential applications in shape matching and mesh abstraction. Our results show that this basis has distinct advantages over existing alternatives, indicating high potential for a wide range of use‐cases in mesh processing.     

Exact Minkowksi Sums of Polyhedra and Exact and Efficient Decomposition of Polyhedra into Convex Pieces

We present the first exact and robust implementation of the 3D Minkowski sum of two non-convex polyhedra. Our implementation decomposes the two polyhedra into convex pieces, performs pairwise Minkowski sums on the convex pieces, and constructs their union. We achieve exactness and the handling of all degeneracies by building upon 3D Nef polyhedra as provided by Cgal. The implementation also supports open and closed polyhedra. This allows the handling of degenerate scenarios like the tight passage problem in robot motion planning. The bottleneck of our approach is the union step. We address efficiency by optimizing this step by two means: we implement an efficient decomposition that yields a small number of convex pieces, and develop, test and optimize multiple strategies for uniting the partial sums by consecutive binary union operations. The decomposition that we implemented as part of the Minkowski sum is interesting in its own right. It is the first robust implementation of a decomposition of polyhedra into convex pieces that yields at most O(r ²) pieces, where r is the number of edges whose adjacent facets comprise an angle of more than 180 degrees with respect to the interior of the polyhedron. This work was partially supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 639.023.301.     

凸组合球面参数化

针对具有单边界的三角网格或与球面同胚的零亏格封闭网格，提出一种基于球面向量线性凸组合的三维网格球面参数化方法．把参数域从平面凸区域扩展到球面凸区域，并把具有凸性的重心坐标纳入到参数化框架中，使得参数化具有保形性质且变形小，同时证明了该参数化方法的存在性和惟一性．整个算法简单可靠．     

Image-space visibility ordering for cell projection volume rendering of unstructured data

Projection methods for volume rendering unstructured data work by projecting, in visibility order, the polyhedral cells of the mesh onto the image plane, and incrementally compositing each cell's color and opacity into the final image. Normally, such methods require an algorithm to determine a visibility order of the cells. The meshed polyhedra visibility order (MPVO) algorithm can provide such an order for convex meshes by considering the implications of local ordering relations between cells sharing a common face. However, in nonconvex meshes, one must also consider ordering relations along viewing rays which cross empty space between cells. In order to include these relations, the algorithm described in this paper, the scanning exact meshed polyhedra visibility ordering (SXMPVO) algorithm, scan-converts the exterior faces of the mesh and saves the ray-face intersections in an A-buffer data structure which is then used for retrieving the extra ordering relations. The image which SXMPVO produces is the same as would be produced by ordering the cells exactly, even though SXMPVO does not compute an exact visibility ordering. This is because the image resolution used for computing the visibility ordering relations is the same as that which is used for the actual volume rendering and we choose our A-buffer rays at the same sample points that are used to establish a polygon's pixel coverage during hardware scan conversion. Thus, the algorithm is image-space correct. The SXMPVO algorithm has several desirable features; among them are speed, simplicity of implementation, and no extra (i.e., with respect to MPVO) preprocessing.     

Mesh Segmentation via Spectral Embedding and Contour Analysis

We propose a mesh segmentation algorithm via recursive bisection where at each step, a sub-mesh embedded in 3D is first spectrally projected into the plane and then a contour is extracted from the planar embedding. We rely on two operators to compute the projection: the well-known graph Laplacian and a geometric operator designed to emphasize concavity. The two embeddings reveal distinctive shape semantics of the 3D model and complement each other in capturing the structural or geometrical aspect of a segmentation. Transforming the shape analysis problem to the 2D domain also facilitates our segmentability analysis and sampling tasks. We propose a novel measure of the segmentability of a shape, which is used as the stopping criterionfor our segmentation. The measure is derived from simple area- and perimeter-based convexity measures. We achieve invariance to shape bending through multi-dimensional scaling (MDS) based on the notion of inner distance. We also utilize inner distances to develop a novel sampling scheme to extract two samples along a contour which correspond to two vertices residing on different parts of the sub-mesh. The two samples are used to derive a spectral linear ordering of the mesh faces. We obtain a final cut via a linear search over the face sequence based on part salience, where a choice of weights for different factors of part salience is guided by the result from segmentability analysis.     

An Efficient Adaptive Algorithm for Constructing the Convex Differences Tree of a Simple Polygon

The convex differences tree (CDT) representation of a simple polygon is useful in computer graphics, computer vision, computer aided design and robotics. The root of the tree contains the convex hull of the polygon and there is a child node recursively representing every connectivity component of the set difference between the convex hull and the polygon. We give an O(n log K + K log² n) time algorithm for constructing the CDT, where n is the number of polygon vertices and K is the number of nodes in the CDT. The algorithm is adaptive to a complexity measure defined on its output while still being worst case efficient. For simply shaped polygons, where K is a constant, the algorithm is linear. In the worst case K = O(n) and the complexity is O(n log² n). We also give an O(n log n) algorithm which is an application of the recently introduced compact interval tree data structure.     

Hierarchical Quad Meshing of 3D Scanned Surfaces

In this paper we present a novel method to reconstruct watertight quad meshes on scanned 3D geometry. There exist many different approaches to acquire 3D information from real world objects and sceneries. Resulting point clouds depict scanned surfaces as sparse sets of positional information. A common downside is the lack of normals, connectivity or topological adjacency data which makes it difficult to actually recover a meaningful surface. The concept described in this paper is designed to reconstruct a surface mesh despite all this missing information. Even when facing varying sample density, our algorithm is still guaranteed to produce watertight manifold meshes featuring quad faces only. The topology can be set‐up to follow superimposed regular structures or align naturally to the point cloud's shape. Our proposed approach is based on an initial divide and conquer subsampling procedure: Surface samples are clustered in meaningful neighborhoods as leafs of a kd‐tree. A representative sample of the surface neighborhood is determined for each leaf using a spherical surface approximation. The hierarchical structure of the binary tree is utilized to construct a basic set of loose tiles and to interconnect them. As a final step, missing parts of the now coherent tile structure are filled up with an incremental algorithm for locally optimal gap closure. Disfigured or concave faces in the resulting mesh can be removed with a constrained smoothing operator.     

Linear Surface Reconstruction from Discrete Fundamental Forms on Triangle Meshes

We present a linear algorithm to reconstruct the vertex coordinates for a surface mesh given its edge lengths and dihedral angles, unique up to rotation and translation. A local integrability condition for the existence of an immersion of the mesh in 3D Euclidean space is provided, mirroring the fundamental theorem of surfaces in the continuous setting (i.e. Gauss's equation and the Mainardi–Codazzi equations) if we regard edge lengths as the discrete first fundamental form and dihedral angles as the discrete second fundamental form. The resulting sparse linear system to solve for the immersion is derived from the convex optimization of a quadratic energy based on a lift from the immersion in the 3D Euclidean space to the 6D rigid motion space. This discrete representation and linear reconstruction can benefit a wide range of geometry processing tasks such as surface deformation and shape analysis. A rotation‐invariant surface deformation through point and orientation constraints is demonstrated as well.     

Semi‐Automatic Conversion of 3D Shape into Flat‐Foldable Polygonal Model

This paper presents a method that can convert a given 3D mesh into a flat‐foldable model consisting of rigid panels. A previous work proposed a method to assist manual design of a single component of such flat‐foldable model, consisting of vertically‐connected side panels as well as horizontal top and bottom panels. Our method semi‐automatically generates a more complicated model that approximates the input mesh with multiple convex components. The user specifies the folding direction of each convex component and the fidelity of shape approximation. Given the user inputs, our method optimizes shapes and positions of panels of each convex component in order to make the whole model flat‐foldable. The user can check a folding animation of the output model. We demonstrate the effectiveness of our method by fabricating physical paper prototypes of flat‐foldable models.     

连接不相交线段成简单多边形(链)的算法及其实现

提出一个如何连接平面上n条线段与一个简单多边形或者简单多边形链的实际问题，并证明了连接平面上线段集S成一简单多边形链的一个充分条件——S中有一条线段连接凸壳CH（S）中不相领顶点。提出了连接平面上线段集S成一简单多边形或者简单多边形链的算法，其基本思想是首先农层计算线段集S的凸壳，并将这些凸壳改变为简单多边形；然后计算各多边形之间的交点，进而删去这些交点；最后俣并若干个简单多边形为一个简单多边形。当S中线段数目n较大时，用分治思想设计分治算法，较好地求解了这个问题。利用计算机求解这个问题具有实际应用价值。     

Fast, Exact, Linear Booleans

We present a new system for robustly performing Boolean operations on linear, 3D polyhedra. Our system is exact, meaning that all internal numeric predicates are exactly decided in the sense of exact geometric computation. Our BSP-tree based system is 16-28× faster at performing iterative computations than CGAL's Nef Polyhedra based system, the current best practice in robust Boolean operations, while being only twice as slow as the non-robust modeler Maya. Meanwhile, we achieve a much smaller substrate of geometric subroutines than previous work, comprised of only 4 predicates, a convex polygon constructor, and a convex polygon splitting routine. The use of a BSP-tree based Boolean algorithm atop this substrate allows us to explicitly handle all geometric degeneracies without treating a large number of cases.     

Modelling 3D shaded solids of arbitrary shape using an edge-oriented algorithm

A fast hidden-line eliminating algorithm is introduced which enables creation of 3D shaded images. The algorithm also supports set operations on bodies. It deals only with solids bounded by planar patches (curved surfaces are approximated by polyhedra). This makes it possible to implement a simple shading technique. Bodies of arbitrary shape, both convex and concave, can be built by using such operations as translation and rotation of a polyline or polygon. The algorithm is designed to be able to run even on personal computers. The CPU time for an IBM AT is in the minute range for scenes composed of 400–500 faces. The program package can thus be used effectively in those cases when shaded images are not very complicated, short processing time is of primary importance and there is no need for pictures of “realistic” quality. Possible fields of application are animation, technical drawings and simulation.     

Skeleton computation of orthogonal polyhedra

Skeletons are powerful geometric abstractions that provide useful representations for a number of geometric operations. The straight skeleton has a lower combinatorial complexity compared with the medial axis. Moreover, while the medial axis of a polyhedron is composed of quadric surfaces the straight skeleton just consist of planar faces. Although there exist several methods to compute the straight skeleton of a polygon, the straight skeleton of polyhedra has been paid much less attention. We require to compute the skeleton of very large datasets storing orthogonal polyhedra. Furthermore, we need to treat geometric degeneracies that usually arise when dealing with orthogonal polyhedra. We present a new approach so as to robustly compute the straight skeleton of orthogonal polyhedra. We follow a geometric technique that works directly with the boundary of an orthogonal polyhedron. Our approach is output sensitive with respect to the number of vertices of the skeleton and solves geometric degeneracies. Unlike the existing straight skeleton algorithms that shrink the object boundary to obtain the skeleton, our algorithm relies on the plane sweep paradigm. The resulting skeleton is only composed of axis‐aligned and 45° rotated planar faces and edges.