1/43 Department of Computer Science and Engineering Delaunay Mesh Generation Tamal K. Dey The Ohio State University.

Slides:



Advertisements
Similar presentations
Department of Computer Science and Engineering Normal Estimation for Point Clouds: A Comparison Study for a Voronoi Based Method Tamal K. DeyGang LiJian.
Advertisements

Alpha Shapes. Used for Shape Modelling Creates shapes out of point sets Gives a hierarchy of shapes. Has been used for detecting pockets in proteins.
BITS Pilani Hyderabad Campus MESH GENERATION Dr. Tathagata Ray Assistant Professor, BITS Pilani, Hyderabad Campus
Surface Reconstruction From Unorganized Point Sets
Discrete Differential Geometry Planar Curves 2D/3D Shape Manipulation, 3D Printing March 13, 2013 Slides from Olga Sorkine, Eitan Grinspun.
Proximity graphs: reconstruction of curves and surfaces
Delaunay Meshing for Piecewise Smooth Complexes Tamal K. Dey The Ohio State U. Joint work: Siu-Wing Cheng, Joshua Levine, Edgar A. Ramos.
Extended Gaussian Images
Sample Shuffling for Quality Hierarchic Surface Meshing.
Flow Complex Joachim Giesen Friedrich-Schiller-Universität Jena.
Medial axis computation of exact curves and surfaces M. Ramanathan Department of Engineering Design, IIT Madras Medial object.
Discrete Geometry Tutorial 2 1
1st Meeting Industrial Geometry Computational Geometry ---- Some Basic Structures 1st IG-Meeting.
Xianfeng Gu, Yaling Wang, Tony Chan, Paul Thompson, Shing-Tung Yau
Computing Stable and Compact Representation of Medial Axis Wenping Wang The University of Hong Kong.
Computing Medial Axis and Curve Skeleton from Voronoi Diagrams Tamal K. Dey Department of Computer Science and Engineering The Ohio State University Joint.
1/50 Department of Computer Science and Engineering Localized Delaunay Refinement for Sampling and Meshing Tamal K. Dey Joshua A. Levine Andrew G. Slatton.
Dual Marching Cubes: An Overview
Discrete geometry Lecture 2 1 © Alexander & Michael Bronstein
2. Voronoi Diagram 2.1 Definiton Given a finite set S of points in the plane , each point X of  defines a subset S X of S consisting of the points of.
1cs542g-term Notes. 2 Meshing goals  Robust: doesn’t fail on reasonable geometry  Efficient: as few triangles as possible Easy to refine later.
UMass Lowell Computer Science Advanced Algorithms Computational Geometry Prof. Karen Daniels Spring, 2007 Chapter 5: Voronoi Diagrams Wednesday,
Tamal K. Dey The Ohio State University Delaunay Meshing of Surfaces.
Surface Reconstruction Some figures by Turk, Curless, Amenta, et al.
OBBTree: A Hierarchical Structure for Rapid Interference Detection Gottschalk, M. C. Lin and D. ManochaM. C. LinD. Manocha Department of Computer Science,
Introductory Notes on Geometric Aspects of Topology PART I: Experiments in Topology 1964 Stephen Barr (with some additional material from Elementary Topology.
reconstruction process, RANSAC, primitive shapes, alpha-shapes
Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.
Voronoi diagrams of “nice” point sets Nina Amenta UC Davis “The World a Jigsaw”
1 University of Denver Department of Mathematics Department of Computer Science.
Tamal K. Dey The Ohio State University Computing Shapes and Their Features from Point Samples.
Anisotropic Voronoi Diagrams and Guaranteed-Quality Anisotropic Mesh Generation François Labelle Jonathan Richard Shewchuk Computer Science Division University.
1/61 Department of Computer Science and Engineering Tamal K. Dey The Ohio State University Delaunay Refinement and Its Localization for Meshing.
Delaunay Triangulations Presented by Glenn Eguchi Computational Geometry October 11, 2001.
Dobrina Boltcheva, Mariette Yvinec, Jean-Daniel Boissonnat INRIA – Sophia Antipolis, France 1. Initialization Use the.
Planning Near-Optimal Corridors amidst Obstacles Ron Wein Jur P. van den Berg (U. Utrecht) Dan Halperin Athens May 2006.
Department of Computer Science and Engineering Practical Algorithm for a Large Class of Domains Tamal K. Dey and Joshua A. Levine The Ohio State University.
Algorithms for Triangulations of a 3D Point Set Géza Kós Computer and Automation Research Institute Hungarian Academy of Sciences Budapest, Kende u
Tamal K. Dey The Ohio State University Computing Shapes and Their Features from Point Samples.
SURFACE RECONSTRUCTION FROM POINT CLOUD Bo Gao Master’s Thesis December, 2007 Thesis Committee: Professor Harriet Fell Professor Robert Futrelle College.
Mesh Generation 58:110 Computer-Aided Engineering Reference: Lecture Notes on Delaunay Mesh Generation, J. Shewchuk (1999)
1 Surface Applications Fitting Manifold Surfaces To 3D Point Clouds, Cindy Grimm, David Laidlaw and Joseph Crisco. Journal of Biomechanical Engineering,
TEL-AVIV UNIVERSITY RAYMOND AND BEVERLY SACKLER FACULTY OF EXACT SCIENCES SCHOOL OF MATHEMATICAL SCIENCES An Algorithm for the Computation of the Metric.
Spectral surface reconstruction Reporter: Lincong Fang 24th Sep, 2008.
Lecture 7 : Point Set Processing Acknowledgement : Prof. Amenta’s slides.
CSE554ContouringSlide 1 CSE 554 Lecture 4: Contouring Fall 2015.
Detecting Undersampling in Surface Reconstruction Tamal K. Dey and Joachim Giesen Ohio State University.
Vertices, Edges and Faces By Jordan Diamond. Vertices In geometry, a vertices is a special kind of point which describes the corners or intersections.
PMR: Point to Mesh Rendering, A Feature-Based Approach Tamal K. Dey and James Hudson
A New Voronoi-based Reconstruction Algorithm
Introductory Notes on Geometric Aspects of Topology PART I: Experiments in Topology 1964 Stephen Barr (with some additional material from Elementary Topology.
UNC Chapel Hill M. C. Lin Delaunay Triangulations Reading: Chapter 9 of the Textbook Driving Applications –Height Interpolation –Constrained Triangulation.
Shape Reconstruction from Samples with Cocone Tamal K. Dey Dept. of CIS Ohio State University.
1/57 CS148: Introduction to Computer Graphics and Imaging Geometric Modeling CS148 Lecture 6.
With Tamal Dey, Qichao Que, Issam Safa, Lei Wang, Yusu Wang Computer science and Engineering The Ohio State University Xiaoyin Ge.
1/66 Department of Computer Science and Engineering Tamal K. Dey The Ohio State University Delaunay Mesh Generation.
CDS 301 Fall, 2008 Domain-Modeling Techniques Chap. 8 November 04, 2008 Jie Zhang Copyright ©
3/3/15CMPS 3130/6130 Computational Geometry1 CMPS 3130/6130 Computational Geometry Spring 2015 Delaunay Triangulations I Carola Wenk Based on: Computational.
Tamal K. Dey The Ohio State University Surface and Volume Meshing with Delaunay Refinement.
Bigyan Ankur Mukherjee University of Utah. Given a set of Points P sampled from a surface Σ,  Find a Surface Σ * that “approximates” Σ  Σ * is generally.
Topology Preserving Edge Contraction Paper By Dr. Tamal Dey et al Presented by Ramakrishnan Kazhiyur-Mannar.
CMPS 3130/6130 Computational Geometry Spring 2017
Decimating Samples for Mesh Simplification
Variational Tetrahedral Meshing
Shape Dimension and Approximation from Samples
Localized Delaunay Refinement For Piecewise-Smooth Complexes
Domain-Modeling Techniques
Delaunay Triangulation & Application
Localized Delaunay Refinement for Volumes
Presentation transcript:

1/43 Department of Computer Science and Engineering Delaunay Mesh Generation Tamal K. Dey The Ohio State University

2/43 Department of Computer Science and Engineering Delaunay Mesh Generation Automatic mesh generation with good quality. Delaunay refinements: The Delaunay triangulation lends to a proof structure. And it naturally optimizes certain geometric properties such as min angle.

3/43 Department of Computer Science and Engineering Input/Output Points P sampled from a surface  in 3D (don’t know   Reconstruct  A simplicial complex K, (i) K has a geometric realization in 3D (ii) |K| homeomorphic to , (iii) Hausdorff distance between |K| and  is small A smooth surface  (or a compact set): Generate a point sample P from  Generate a simplicial complex K with vert K=P and satisfying (i), (ii), (iii).

4/43 Department of Computer Science and Engineering Surface Reconstruction ` Point Cloud Surface Reconstruction

5/43 Department of Computer Science and Engineering Medial Axis

6/43 Department of Computer Science and Engineering Local Feature Size (Smooth) Local feature size is calculated using the medial axis of a smooth shape. f(x) is the distance from a point to the medial axis

7/43 Department of Computer Science and Engineering Each x has a sample within  f(x) distance  -Sample [ABE98] x

8/43 Department of Computer Science and Engineering Voronoi/Delaunay

9/43 Department of Computer Science and Engineering Normal and Voronoi Cells(3D) [Amenta-Bern SoCG98]

10/43 Department of Computer Science and Engineering Poles P+P+ P-P-

11/43 Department of Computer Science and Engineering Normal Lemma The angle between the pole vector v p and the normal n p is O(  ). P+P+ P-P- npnp vpvp

12/43 Department of Computer Science and Engineering Restricted Delaunay If the point set is sampled from a domain D. We can define the restricted Delaunay triangulation, denoted Del P| D. Each simplex   Del P| D is the dual of a Voronoi face V  that has a nonempty intersection with the domain D.

13/43 Department of Computer Science and Engineering Topological Ball Property (TBP) P has the TBP for a manifold  if each k- face in Vor P either does not intersect  or intersects in a topological (k-1)-ball. Thm (Edelsbrunner- Shah97 ) If P has the TBP then Del P|  is homeomorphic to .

14/43 Department of Computer Science and Engineering Cocone (Amenta-Choi-D.-Leekha) v p = p + - p is the pole vector Space spanned by vectors within the Voronoi cell making angle > 3  /8 with v p or -v p

15/43 Department of Computer Science and Engineering Cocone Algorithm

16/43 Department of Computer Science and Engineering Cocone Guarantees Theorem: Any point x   is within O(  f(x) distance from a point in the output. Conversely, any point of output surface has a point x   within O(  )f(x) distance. Triangle normals make O(  ) angle with true normals at vertices. Theorem: The output surface computed by Cocone from an  -sample is homeomorphic to the sampled surface for sufficiently small .

17/43 Department of Computer Science and Engineering Meshing Input Polyhedra Smooth Surfaces Piecewise-smooth Surfaces Non-manifolds &

18/43 Department of Computer Science and Engineering Basics of Delaunay Refinement Pioneered by Chew89, Ruppert92, Shewchuck98 To mesh some domain D, 1. Initialize a set of points P  D, compute Del P. 2. If some condition is not satisfied, insert a point c from D into P and repeat step Return Del P| D. Burden is to show that the algorithm terminates (shown by a packing argument).

19/43 Department of Computer Science and Engineering Polyhedral Meshing Output mesh conforms to input: All input edges meshed as a collection of Delaunay edges. All input facets are meshed with a collection of Delaunay triangles. Algorithms with angle restrictions: Chew89, Ruppert92, Miller-Talmor- Teng-Walkington95, Shewchuk98. Small angles allowed: Shewchuk00, Cohen-Steiner- Verdiere-Yvinec02, Cheng-Poon03, Cheng-Dey-Ramos-Ray04, Pav- Walkington04.

20/43 Department of Computer Science and Engineering Local Feature Size (Polyhedral) g(x) = the radius of the smallest ball placed at x which intersects the domain in two disjoint elements pieces. g(x) is Lipschitz, |g(x) - g(y)| <= |x - y|. Termination for polyhedral meshing is shown by a packing argument using this local feature size.

21/43 Department of Computer Science and Engineering Smooth Surface Meshing Input mesh is either an implicit surface or a polygonal mesh approximating a smooth surface Output mesh approximates input geometry, conforms to input topology: No guarantees: Chew93. Skin surfaces: Cheng-Dey-Edelsbrunner- Sullivan01. Provable surface algorithms: Boissonnat-Oudot03 and Cheng- Dey-Ramos-Ray04. Interior Volumes: Oudot-Rineau-Yvinec06.

22/43 Department of Computer Science and Engineering Homeomorphism and Isotopy Homeomorphsim: A function f between two topological spaces: f is a bijection f and f -1 are both continuous Isotopy: A continuous deformation maintaining homeomorphism  

23/43 Department of Computer Science and Engineering Sampling Theorem Theorem (Boissonat-Oudot 2005): If P   is a discrete sample of a smooth surface  so that each x where a Voronoi edge intersects  lies within  f(x) distance from a sample, then for  <0.09, the restricted Delaunay triangulation Del P|   has the following properties: (i)It is homeomorphic to  (even isotopic embeddings). (ii)Each triangle has normal aligning within O(  ) angle to the surface normals (iii)Hausdorff distance between  and Del P|   is O(   ) of the local feature size. Theorem :(Amenta-Bern 98, Cheng-Dey-Edelsbrunner-Sullivan 01) If P   is a discrete   sample of a smooth surface  then for  < 0.09 the restricted Delaunay triangulation Del P|   has  the following properties: Sampling Theorem Modified

24/43 Department of Computer Science and Engineering Basic Delaunay Refinement 1. Initialize a set of points P  , compute Del P. 2. If some condition is not satisfied, insert a point c from  into P and repeat step Return Del P| . Surface Delaunay Refinement 2. If some Voronoi edge intersects  at x with d(x,P)>  f(x) insert x in P.

25/43 Department of Computer Science and Engineering Difficulty How to compute f(x)? Special surfaces such as skin surfaces allow easy computation of f(x) [CDES01] Can be approximated by computing approximate medial axis, needs a dense sample.

26/43 Department of Computer Science and Engineering A Solution Replace d(x,P)<  f(x) with d(x,P)< an user parameter But, this does not guarantee any topology Require that triangles around vertices form topological disks Guarantees that output is a manifold

27/43 Department of Computer Science and Engineering A Solution 1. Initialize a set of points P  , compute Del P. 2. If some Voronoi edge intersects  at x with d(x,P)>  f(x) insert x in P, and repeat step (b)If restricted triangles around a vertex p do not form a topological disk, insert furthest x where a dual Voronoi edge of a triangle around p intersects . 3. Return Del P|  2. (a) If some Voronoi edge intersects  at x with d(x,P)> insert x in P, and repeat step 2(a). Algorithm DelSurf( , ) X=center of largest Surface Delaunay ball x

28/43 Department of Computer Science and Engineering A MeshingTheorem Theorem: The algorithm DelSurf produces output mesh with the following guarantees: (i)The output mesh is always a 2-manifold (ii)If  is sufficiently small, the output mesh  satisfies topological and geometric guarantees: 1.It is related to  with an isotopy  2.Each triangle has normal aligning within O( ) angle to the surface normals 3.Hausdorff distance between  and Del P|   is O(  ) of the local feature size.

29/43 Department of Computer Science and Engineering Implicit surface

30/43 Department of Computer Science and Engineering Remeshing

31/43 Department of Computer Science and Engineering PSCs – A Large Input Class [Cheng-D.-Ramos 07] Piecewise smooth complexes (PSCs) include: Polyhedra Smooth Surfaces Piecewise-smooth Surfaces Non-manifolds &

32/43 Department of Computer Science and Engineering Protecting Ridges 1. Balls must cover each element of D 1 completely. 2. Any 2 adjacent balls on a 1-face must overlap significantly without containing each others centers. 3. No 3 balls should have a common intersection. 4. (Tangent/Normal Variation) For any point p on a curve, if we look in a small enough region 1. The portion of the curve nearby p is a single piece. 2. The tangent along this piece varies a small amount. 3. The normal of each surface piece adjacent to p also varies little.

33/43 Department of Computer Science and Engineering Protecting Ridges

34/43 Department of Computer Science and Engineering A New Disk Condition Cheng-Dey-Levine use a single topological disk condition: For a point p on a 2-face σ, Umb D (p) is the set of triangles incident to p, restricted to D. Umb σ (p) is the set of triangles incident to p, restricted to σ. DiskCondition(p) requires: i. Umb D (p) =  σ, p  σ Umb σ (p) ii. For each σ containing p, Umb σ (p) is a 2-disk where p is in the interior iff p  int σ DiskCondition() satisfied

35/43 Department of Computer Science and Engineering A New Disk Condition Cheng-Dey-Levine use a single topological disk condition: For a point p on a 2-face σ, Umb D (p) is the set of triangles incident to p, restricted to D. Umb σ (p) is the set of triangles incident to p, restricted to σ. DiskCondition(p) requires: i. Umb D (p) =  σ, p  σ Umb σ (p) ii. For each σ containing p, Umb σ (p) is a 2-disk where p is in the interior iff p  int σ DiskCondition() satisfied

36/43 Department of Computer Science and Engineering DelPSC Algorithm [Cheng-D.-Ramos-Levine 07,08] DelPSC(D, λ) 1. Protect ridges of D using protection balls. 2. Refine in the weighted Delaunay by turning the balls into weighted points. 1. Refine a triangle if it has orthoradius > l. 2. Refine a triangle or a ball if disk condition is violated 3. Refine a ball if it is too big. 3. Return  i Del i S| Di

37/43 Department of Computer Science and Engineering Guarantees for DelPSC 1. Manifold For each σ  D 2, triangles in Del S| σ are a manifold with vertices only in σ. Further, their boundary is homeomorphic to bd σ with vertices only in σ. 2. Granularity There exists some λ > 0 so that the output of DelPSC(D, λ) is homeomorphic to D. This homeomorphism respects stratification, For 0 ≤ i ≤ 2, and σ  D i, Del S| σ is homemorphic to σ too.

38/43 Department of Computer Science and Engineering Reducing λ

39/43 Department of Computer Science and Engineering Examples

40/43 Department of Computer Science and Engineering Examples

41/43 Department of Computer Science and Engineering Examples

42/43 Department of Computer Science and Engineering Examples

43/43 Department of Computer Science and Engineering Some Resources Software available from h ttp:// Open : Reconstruct piecewise smooth surfaces, non-manifolds Open: Guarantee quality of all tetrahedra in volume meshing A book Delaunay Mesh Generation: w/ S.-W. Cheng, J. Shewchuk (2012)

44/43 Department of Computer Science and Engineering Thank You!