An Approach of Using Delaunay Refinement to Mesh Continuous Height Fields
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Computing Three-Dimensional Constrained Delaunay Refinement Using the GPU
Computing Three-dimensional Constrained Delaunay Refinement Using the GPU Zhenghai Chen Tiow-Seng Tan School of Computing School of Computing National University of Singapore National University of Singapore [email protected] [email protected] ABSTRACT collectively called elements, are split in this order in many rounds We propose the first GPU algorithm for the 3D triangulation refine- until there are no more bad tetrahedra. Each round, the splitting ment problem. For an input of a piecewise linear complex G and is done to many elements in parallel with many GPU threads. The a constant B, it produces, by adding Steiner points, a constrained algorithm first calculates the so-called splitting points that can split Delaunay triangulation conforming to G and containing tetrahedra elements into smaller ones, then decides on a subset of them to mostly of radius-edge ratios smaller than B. Our implementation of be Steiner points for actual insertions into the triangulation T . the algorithm shows that it can be an order of magnitude faster than Note first that a splitting point is calculated by a GPU threadas the best CPU algorithm while using a similar amount of Steiner the midpoint of a subsegment, the circumcenter of the circumcircle points to produce triangulations of comparable quality. of the subface, and the circumcenter of the circumsphere of the tetrahedron. Note second that not all splitting points calculated CCS CONCEPTS can be inserted as Steiner points in a same round as they together can potentially create undesirable short edges in T to cause non- • Theory of computation → Computational geometry; • Com- termination of the algorithm. -
Marching Cubes and Histogram Pyramids for 3D Medical Visualization
Journal of Imaging Article Marching Cubes and Histogram Pyramids for 3D Medical Visualization Porawat Visutsak Department of Computer and Information Science, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand; [email protected]; Tel.: +66-2555-2000 (ext. 4225) Received: 1 July 2020; Accepted: 31 August 2020; Published: 3 September 2020 Abstract: This paper aims to implement histogram pyramids with marching cubes method for 3D medical volumetric rendering. The histogram pyramids are used for feature extraction by segmenting the image into the hierarchical order like the pyramid shape. The histogram pyramids can decrease the number of sparse matrixes that will occur during voxel manipulation. The important feature of the histogram pyramids is the direction of segments in the image. Then this feature will be used for connecting pixels (2D) to form up voxel (3D) during marching cubes implementation. The proposed method is fast and easy to implement and it also produces a smooth result (compared to the traditional marching cubes technique). The experimental results show the time consuming for generating 3D model can be reduced by 15.59% in average. The paper also shows the comparison between the surface rendering using the traditional marching cubes and the marching cubes with histogram pyramids. Therefore, for the volumetric rendering such as 3D medical models and terrains where a large number of lookups in 3D grids are performed, this method is a particularly good choice for generating the smooth surface of 3D object. Keywords: marching cubes; histogram pyramids; volumetric rendering; 3D medical model; smooth voxel; isosurface 1. -
Efficient Mesh Optimization Schemes Based on Optimal Delaunay
Comput. Methods Appl. Mech. Engrg. 200 (2011) 967–984 Contents lists available at ScienceDirect Comput. Methods Appl. Mech. Engrg. journal homepage: www.elsevier.com/locate/cma Efficient mesh optimization schemes based on Optimal Delaunay Triangulations q ⇑ Long Chen a, , Michael Holst b a Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA b Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, USA article info abstract Article history: In this paper, several mesh optimization schemes based on Optimal Delaunay Triangulations are devel- Received 15 April 2010 oped. High-quality meshes are obtained by minimizing the interpolation error in the weighted L1 norm. Received in revised form 2 November 2010 Our schemes are divided into classes of local and global schemes. For local schemes, several old and new Accepted 10 November 2010 schemes, known as mesh smoothing, are derived from our approach. For global schemes, a graph Lapla- Available online 16 November 2010 cian is used in a modified Newton iteration to speed up the local approach. Our work provides a math- ematical foundation for a number of mesh smoothing schemes often used in practice, and leads to a new Keywords: global mesh optimization scheme. Numerical experiments indicate that our methods can produce well- Mesh smoothing shaped triangulations in a robust and efficient way. Mesh optimization Mesh generation Ó 2010 Elsevier B.V. All rights reserved. Delaunay triangulation Optimal Delaunay Triangulation 1. Introduction widely used harmonic energy in moving mesh methods [8–10], summation of weighted edge lengths [11,12], and the distortion We shall develop fast and efficient mesh optimization schemes energy used in the approach of Centroid Voronoi Tessellation based on Optimal Delaunay Triangulations (ODTs) [1–3]. -
A Generic Framework for Delaunay Mesh Generation Clément Jamin, Pierre Alliez, Mariette Yvinec, Jean-Daniel Boissonnat
CGALmesh: a Generic Framework for Delaunay Mesh Generation Clément Jamin, Pierre Alliez, Mariette Yvinec, Jean-Daniel Boissonnat To cite this version: Clément Jamin, Pierre Alliez, Mariette Yvinec, Jean-Daniel Boissonnat. CGALmesh: a Generic Framework for Delaunay Mesh Generation. ACM Transactions on Mathematical Software, Association for Computing Machinery, 2015, 41 (4), pp.24. 10.1145/2699463. hal-01071759 HAL Id: hal-01071759 https://hal.inria.fr/hal-01071759 Submitted on 6 Oct 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 1 CGALmesh: a Generic Framework for Delaunay Mesh Generation CLEMENT JAMIN, Inria and Universite´ Lyon 1, LIRIS, UMR5205 PIERRE ALLIEZ, Inria MARIETTE YVINEC, Inria JEAN-DANIEL BOISSONNAT, Inria CGALmesh is the mesh generation software package of the Computational Geometry Algorithm Library (CGAL). It generates isotropic simplicial meshes – surface triangular meshes or volume tetrahedral meshes – from input surfaces, 3D domains as well as 3D multi-domains, with or without sharp features. The under- lying meshing algorithm relies on restricted Delaunay triangulations to approximate domains and surfaces, and on Delaunay refinement to ensure both approximation accuracy and mesh quality. CGALmesh provides guarantees on approximation quality as well as on the size and shape of the mesh elements. -
Revisiting Delaunay Refinement Triangular Mesh Generation
Revisiting Delaunay Refinement Triangular Mesh Generation on Curve-bounded Domains Serge Gosselin and Carl Ollivier-Gooch Department of Mechanical Engineering, University of British Columbia, Vancouver, BC, V6T 1Z4, Canada Email: [email protected] ABSTRACT gorithms eliminate small angles from a triangulation, thus improving its quality. The first practically rele- An extension of Shewchuk’s Delaunay Refinement al- vant algorithms are due to Ruppert [9] and Chew [3]. gorithm to planar domains bounded by curves is pre- These not only produce meshes of provably good qual- sented. A novel method is applied to construct con- ity, but also provide theoretical guarantees regarding strained Delaunay triangulations from such geome- mesh sizing and, under certain input conditions, termi- tries. The quality of these triangulations is then im- nation. Ruppert’s and Chew’s schemes are sufficiently proved by inserting vertices at carefully chosen lo- similar for Shewchuk to analyze them in a common cations. Amendments to Shewchuk’s insertion rules framework [10, 13]. His findings lead to an hybrid al- are necessary to handle cases resulting from the pres- gorithm improving quality guarantees and relaxing in- ence of curves. The algorithm accepts small input an- put restrictions. Shewchuk’s algorithm is presented in gles, although special provisions must be taken in their section 3. neighborhood. In practice, our algorithm generates graded triangular meshes where no angle is less than The aforementioned algorithms have the common 30◦, except near small input angles. Such meshes are shortcoming of only accepting input domains whose suitable for use in numerical simulations. boundaries are piecewise linear. This restriction can be impractical in applications where geometric infor- mation comes, for instance, in the form of parametric 1 INTRODUCTION curves. -
Delaunay Refinement Algorithms for Numerical Methods
Delaunay Refinement Algorithms for Numerical Methods Alexander Rand April 30, 2009 Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213 Thesis Committee: Noel Walkington, Chair Shlomo Ta’asan Giovanni Leoni Gary Miller Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Copyright c 2009 Alexander Rand Keywords: mesh generation, Delaunay refinement, finite element interpolation Abstract Ruppert’s algorithm is an elegant solution to the mesh generation problem for non-acute domains in two dimensions. This thesis develops a three-dimensional Delaunay refinement algorithm which produces a conforming Delaunay tetrahedral- ization, ensures a bound on the radius-edge ratio of nearby all tetrahedra, generates tetrahedra of a size related to the local feature size and the size of nearby small in- put angles, and is simple enough to admit an implementation. To do this, Delaunay refinement algorithms for estimating local feature size are constructed. These es- timates are then used to determine an appropriately sized protection region around acutely adjacent features of the input. Finally, a simple variant of Ruppert’s algo- rithm can be applied to produce a quality mesh. Additionally, some finite element interpolation results pertaining to Delaunay refinement algorithms in two dimensions are considered. iv Acknowledgments My work has been supported by many people of whom a number will surely be omitted. First and foremost, I acknowledge the years of support from my family: my parents, my grandparents, and my sister. They have truly shaped who I am today. Many of my undergraduate professors encouraged my decision to pursue further studies in mathematics. -
Deforming Meshes That Split and Merge
Deforming Meshes that Split and Merge Chris Wojtan Nils Thurey¨ Markus Gross Greg Turk Georgia Institute of Technology ETH Zurich¨ ETH Zurich¨ Georgia Institute of Technology Figure 1: Dropping viscoelastic balls in an Eulerian fluid simulation. Invisible geometry is quickly deleted, while the visible surfaces retain their details even after translating through the air and splashing on the ground. Abstract order to produce plausible animations of these deforming materials, we must develop simulation techniques that retain visual details, but We present a method for accurately tracking the moving surface of at the same time allow topological changes to the surface. This is deformable materials in a manner that gracefully handles topologi- the focus of our work. cal changes. We employ a Lagrangian surface tracking method, and We introduce a method of tracking and updating the surface of a we use a triangle mesh for our surface representation so that fine deforming object in a manner that gracefully allows for topology features can be retained. We make topological changes to the mesh changes. Our approach uses a mesh to represent the surface of ob- by first identifying merging or splitting events at a particular grid jects. The key benefit of using a mesh is that it allows the surface to resolution, and then locally creating new pieces of the mesh in the be moved in a Lagrangian manner, using a high-order ODE integra- affected cells using a standard isosurface creation method. We stitch tor to move the vertices of the surface. This results in the retention the new, topologically simplified portion of the mesh to the rest of of fine surface details. -
Lecture Notes on Delaunay Mesh Generation
Lecture Notes on Delaunay Mesh Generation Jonathan Richard Shewchuk February 5, 2012 Department of Electrical Engineering and Computer Sciences University of California at Berkeley Berkeley, CA 94720 Copyright 1997, 2012 Jonathan Richard Shewchuk Supported in part by the National Science Foundation under Awards CMS-9318163, ACI-9875170, CMS-9980063, CCR-0204377, CCF-0430065, CCF-0635381, and IIS-0915462, in part by the University of California Lab Fees Research Program, in part bythe Advanced Research Projects Agency and Rome Laboratory, Air Force Materiel Command, USAF under agreement number F30602- 96-1-0287, in part by the Natural Sciences and Engineering Research Council of Canada under a 1967 Science and Engineering Scholarship, in part by gifts from the Okawa Foundation and the Intel Corporation, and in part by an Alfred P. Sloan Research Fellowship. Keywords: mesh generation, Delaunay refinement, Delaunay triangulation, computational geometry Contents 1Introduction 1 1.1 Meshes and the Goals of Mesh Generation . ............ 3 1.1.1 Domain Conformity . .... 4 1.1.2 ElementQuality ................................ .... 5 1.2 A Brief History of Mesh Generation . ........... 7 1.3 Simplices, Complexes, and Polyhedra . ............. 12 1.4 Metric Space Topology . ........ 15 1.5 HowtoMeasureanElement ........................... ....... 17 1.6 Maps and Homeomorphisms . ....... 21 1.7 Manifolds....................................... ..... 22 2Two-DimensionalDelaunayTriangulations 25 2.1 Triangulations of a Planar Point Set . ............. 26 2.2 The Delaunay Triangulation . .......... 26 2.3 The Parabolic Lifting Map . ......... 28 2.4 TheDelaunayLemma................................ ...... 30 2.5 The Flip Algorithm . ....... 32 2.6 The Optimality of the Delaunay Triangulation . ................ 34 2.7 The Uniqueness of the Delaunay Triangulation . ............... 35 2.8 Constrained Delaunay Triangulations in the Plane . -
Adaptive Surface Extraction from Anisotropic Volumetric Data: Contouring on Generalized Octrees Ricardo Uribe Lobello, Florence Denis, Florent Dupont
Adaptive surface extraction from anisotropic volumetric data: contouring on generalized octrees Ricardo Uribe Lobello, Florence Denis, Florent Dupont To cite this version: Ricardo Uribe Lobello, Florence Denis, Florent Dupont. Adaptive surface extraction from anisotropic volumetric data: contouring on generalized octrees. Annals of Telecommunications - annales des télé- communications, Springer, 2014, 69 (5-6), pp.331-343. 10.1007/s12243-013-0369-4. hal-01119413 HAL Id: hal-01119413 https://hal.archives-ouvertes.fr/hal-01119413 Submitted on 27 Mar 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Noname manuscript No. (will be inserted by the editor) Adaptive surface extraction from anisotropic volumetric data: contouring on generalized octrees Ricardo Uribe Lobello, Florence Denis and Florent Dupont the date of receipt and acceptance should be inserted later Abstract In this article, we present an algorithm to larly, the recent developments of time-of-flight camera extract adaptive surfaces from anisotropic volumetric technology makes it possible to create 3D models from data. For example, this kind of data can be obtained objects by using depth information in a scene. from a set of segmented images, from the sampling of Volumetric datasets are often characterized by a strong an implicit function or it can be built by using depth im- anisotropy caused by the increasing resolution of images ages produced by time-of-flight cameras. -
Decoupling Method for Parallel Delaunay Two-Dimensional Mesh Generation
W&M ScholarWorks Dissertations, Theses, and Masters Projects Theses, Dissertations, & Master Projects 2007 Decoupling method for parallel Delaunay two-dimensional mesh generation Leonidas Linardakis College of William & Mary - Arts & Sciences Follow this and additional works at: https://scholarworks.wm.edu/etd Part of the Computer Sciences Commons Recommended Citation Linardakis, Leonidas, "Decoupling method for parallel Delaunay two-dimensional mesh generation" (2007). Dissertations, Theses, and Masters Projects. Paper 1539623520. https://dx.doi.org/doi:10.21220/s2-mqsk-5d79 This Dissertation is brought to you for free and open access by the Theses, Dissertations, & Master Projects at W&M ScholarWorks. It has been accepted for inclusion in Dissertations, Theses, and Masters Projects by an authorized administrator of W&M ScholarWorks. For more information, please contact [email protected]. Decoupling Method for Parallel Delaunay 2D Mesh Generation Leonidas Linardakis Athens Greece M.Sc. C.S. College of William & M ary M.Sc. Mathematics University of Ioannina A Dissertation presented to the Graduate Faculty of the College of William and Mary in Candidacy for the Degree of Doctor of Philosophy Department of Computer Science The College of William and Mary August 2007 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Copyright 2007 Leonidas Linardakis Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPROVAL PAGE This Dissertation -
Template-Based Remeshing for Image Decomposition
Template-based Remeshing for Image Decomposition Mario A. S. Lizier Marcelo F. Siqueira Joel Daniels II Claudio T. Silva Luis G. Nonato ICMC-USP DIMAp-UFRN SCI-Utah SCI-Utah ICMC-USP Sao˜ Carlos, Brazil Natal, Brazil Salt Lake City, USA Salt Lake City, USA Sao˜ Carlos, Brazil [email protected] [email protected] [email protected] [email protected] [email protected] Abstract—This paper presents a novel quad-based remesh- from a description of the domain is intrinsically harder than ing scheme, which can be regarded as a replacement for generating triangle meshes. Yet, quadrilateral meshes are the triangle-based improvement step of an existing image- more appropriate than triangle meshes for some numerical based mesh generation technique called Imesh. This remesh- ing scheme makes it possible for the algorithm to gener- methods and simulations [7]. ate good quality quadrilateral meshes directly from imaging Contributions. We describe an extension of Imesh which data. The extended algorithm combines two ingredients: (1) generates quadrilateral meshes directly from imaging data. a template-based triangulation-to-quadrangulation conversion strategy and (2) an optimization-based smoothing procedure. Our extension combines two ingredients: a template-based Examples of meshes generated by the extended algorithm, and triangulation-to-quadrangulation conversion strategy, and an an evaluation of the quality of those meshes are presented as optimization-based smoothing procedure. The former aims well. at generating quadrilateral meshes that respect the image Keywords-Image-based mesh generation; template-based object boundaries (as defined by the mesh partition step of mesh; remeshing; mesh smoothing; mesh segmentation; Bezier´ Imesh). -
Tetgen a Quality Tetrahedral Mesh Generator and Three-Dimensional Delaunay Triangulator
TetGen A Quality Tetrahedral Mesh Generator and Three-Dimensional Delaunay Triangulator Version 1.4 User’s Manual January 18, 2006 Hang Si [email protected] http://tetgen.berlios.de c 2002, 2004, 2005, 2006 Abstract TetGen generates tetrahedral meshes and Delaunay tetrahedral- izations. The tetrahedral meshes are suitable for finite element and finite volume methods. The algorithms implemented are the state of the art. This documents briefly explains the problems solved by TetGen and is a detailed user’s guide. Readers will learn how to create tetra- hedral meshes using input files from the command line. Furthermore, the programming interface for calling TetGen from other programs is explained. keywords: tetrahedral mesh, Delaunay tetrahedralization, con- strained Delaunay tetrahedralization, mesh quality, mesh generation 2 Contents 1 Introduction 5 1.1DelaunayTriangulationandTheConvexHull......... 5 1.2ConstrainedDelaunayTetrahedralization............ 8 1.2.1 Piecewise Linear Complex . ............. 12 1.3QualityTetrahedralMesh.................... 13 1.3.1 TheRadius-EdgeRatioQualityMeasure........ 14 2 Getting Started 16 2.1Compilation............................ 16 2.1.1 Unix/Linux........................ 16 2.1.2 Windows 9x/NT/2000/XP . ............. 17 2.2Testing............................... 17 2.3Visualization........................... 20 2.3.1 TetView.......................... 20 2.3.2 OtherMeshViewers................... 21 3UsingTetGen 22 3.1CommandLineSyntax...................... 22 3.2CommandLineSwitches..................... 22