Planar Minimization Diagrams via Subdivision with Applications to Anisotropic Voronoi Diagrams
Let X = {f1, …, fn} be a set of scalar functions of the form fi : ℝ2 → ℝ which satisfy some natural properties. We describe a subdivision algorithm for computing a clustered ε‐isotopic approximation of the minimization diagram of X. By exploiting soft predicates and clustering of Voronoi vertices, o...
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| Vydáno v: | Computer graphics forum Ročník 35; číslo 5; s. 229 - 247 |
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01.08.2016
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| Abstract | Let X = {f1, …, fn} be a set of scalar functions of the form fi : ℝ2 → ℝ which satisfy some natural properties. We describe a subdivision algorithm for computing a clustered ε‐isotopic approximation of the minimization diagram of X. By exploiting soft predicates and clustering of Voronoi vertices, our algorithm is the first that can handle arbitrary degeneracies in X, and allow scalar functions which are piecewise smooth, and not necessarily semi‐algebraic.
We apply these ideas to the computation of anisotropic Voronoi diagram of polygonal sets; this is a natural generalization of anisotropic Voronoi diagrams of point sites, which extends multiplicatively weighted Voronoi diagrams. We implement a prototype of our anisotropic algorithm and provide experimental results. |
|---|---|
| AbstractList | Let X = {f
1
, …, f
n
} be a set of scalar functions of the form f
i
: ℝ
2
→ ℝ which satisfy some natural properties. We describe a subdivision algorithm for computing a clustered ε‐isotopic approximation of the minimization diagram of X. By exploiting soft predicates and clustering of Voronoi vertices, our algorithm is the first that can handle arbitrary degeneracies in X, and allow scalar functions which are piecewise smooth, and not necessarily semi‐algebraic.
We apply these ideas to the computation of anisotropic Voronoi diagram of polygonal sets; this is a natural generalization of anisotropic Voronoi diagrams of point sites, which extends multiplicatively weighted Voronoi diagrams. We implement a prototype of our anisotropic algorithm and provide experimental results. Let X = {f1, …, fn} be a set of scalar functions of the form fi : ℝ2 → ℝ which satisfy some natural properties. We describe a subdivision algorithm for computing a clustered ε‐isotopic approximation of the minimization diagram of X. By exploiting soft predicates and clustering of Voronoi vertices, our algorithm is the first that can handle arbitrary degeneracies in X, and allow scalar functions which are piecewise smooth, and not necessarily semi‐algebraic. We apply these ideas to the computation of anisotropic Voronoi diagram of polygonal sets; this is a natural generalization of anisotropic Voronoi diagrams of point sites, which extends multiplicatively weighted Voronoi diagrams. We implement a prototype of our anisotropic algorithm and provide experimental results. Let X = {f1, ..., fn} be a set of scalar functions of the form fi : 2 [arrow right] which satisfy some natural properties. We describe a subdivision algorithm for computing a clustered [epsi]-isotopic approximation of the minimization diagram of X. By exploiting soft predicates and clustering of Voronoi vertices, our algorithm is the first that can handle arbitrary degeneracies in X, and allow scalar functions which are piecewise smooth, and not necessarily semi-algebraic. We apply these ideas to the computation of anisotropic Voronoi diagram of polygonal sets; this is a natural generalization of anisotropic Voronoi diagrams of point sites, which extends multiplicatively weighted Voronoi diagrams. We implement a prototype of our anisotropic algorithm and provide experimental results. Let X = {f sub(1), ..., f sub(n)} be a set of scalar functions of the form f sub(i) : super(2) arrow right which satisfy some natural properties. We describe a subdivision algorithm for computing a clustered epsilon -isotopic approximation of the minimization diagram of X. By exploiting soft predicates and clustering of Voronoi vertices, our algorithm is the first that can handle arbitrary degeneracies in X, and allow scalar functions which are piecewise smooth, and not necessarily semi-algebraic. We apply these ideas to the computation of anisotropic Voronoi diagram of polygonal sets; this is a natural generalization of anisotropic Voronoi diagrams of point sites, which extends multiplicatively weighted Voronoi diagrams. We implement a prototype of our anisotropic algorithm and provide experimental results. |
| Author | Yap, C. Papadopoulou, E. Bennett, H. |
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| Copyright | 2016 The Author(s) Computer Graphics Forum © 2016 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. 2016 The Eurographics Association and John Wiley & Sons Ltd. |
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| Notes | ArticleID:CGF12979 istex:FE3E80C9E7922B52CEDE558E09766C85AE01521A ark:/67375/WNG-HG73N5NS-8 Chee and Huck are supported by NSF Grant #CCF‐1423228. Evanthia is supported by SNSF #20GG21‐134355, #200021E‐154387. SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-1 ObjectType-Feature-2 content type line 23 |
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| References_xml | – reference: Samet H.: The Design and Analysis of Spatial Data Structures. Addison Wesley, 1990. 2 – reference: Okabe A., Boots B., Sugihara K., Chiu S.N.: Spatial Tessellations - Concepts and Applications of Voronoi Diagrams, 2nd ed. John Wiley and Sons, 2000. 1 – reference: Ratschek H., Rokne J.: Computer Methods for the Range of Functions. Horwood Publishing Limited, Chichester, West Sussex, UK, 1984. 5 – reference: Moore R.E., Kioustelidis J.B.: A simple test for accuracy of approximate solutions to nonlinear (or linear) systems. SIAM J. Numer. Anal. 17, 4 (1980), 521-529. 6, 15, 16 – reference: Wang C., Chiang Y.-J., Yap C.: On Soft Predicates in Subdivision Motion Planning. Comput. Geometry: Theory and Appl. 48, 8 (Sept. 2015), 589-605. doi: 10.1016/j.comgeo.2015.04.002. 2 – reference: Edelsbrunner H., Seidel R.: Voronoi diagrams and arrangements. Discrete and Comp. Geom. 1 (1986), 25-44. 2 – reference: Lin L., Yap C.: Adaptive isotopic approximation of nonsingular curves: the parameterizability and nonlocal isotopy approach. Discrete and Comp. Geom. 45, 4 (2011), 760-795. 14 – reference: Yap C.K.: An O (nlog n) algorithm for the Voronoi diagram for a set of simple curve segments. Discrete and Comp. Geom. 2 (1987), 365-394. 1 – reference: Munkres J.R.: Elements of Algebraic Topology. The Benjamin/Cummings Publishing Company, Inc, Menlo Park, CA, 1984. 2 – reference: Burr M., Choi S., Galehouse B., Yap C.: Complete subdivision algorithms, II: Isotopic meshing of singular algebraic curves. J. Symbolic Computation 47, 2 (2012), 131-152. Special Issue for ISSAC 2008. 3, 6, 14, 15 – reference: Emiris I., Mantzaflaris A., Mourrain B.: Voronoi Diagrams of algebraic distance fields. Computer Aided Design 45, 2 (2013), 511-516. 2, 4, 9 – reference: Chang E.-C., Choi S.W., Kwon D., Park H., Yap C.: Shortest paths for disc obstacles is computable. Int'l. J. Comput. Geometry and Appl. 16, 5-6 (2006), 567-590. Special Issue of IJCGA on Geometric Constraints. (Eds. X.S. Gao and D. Michelucci). 2, 3 – reference: Aurenhammer F., Klein R., Lee D.-T.: Voronoi Diagrams and Delaunay Triangulations. World Scientific, 2013. 1 – volume: 16 start-page: 567 issue: 5 year: 2006 end-page: 6 590 article-title: Shortest paths for disc obstacles is computable publication-title: Int'l. J. Comput. Geometry and Appl. – volume: 47 start-page: 131 issue: 2 year: 2012 end-page: 152 article-title: Complete subdivision algorithms, II: Isotopic meshing of singular algebraic curves publication-title: J. Symbolic Computation – volume: 1 start-page: 25 year: 1986 end-page: 44 article-title: Voronoi diagrams and arrangements publication-title: Discrete and Comp. Geom. – start-page: 191 year: 2003 end-page: 200 – volume: 7921 start-page: 434 year: 2013 end-page: 444 – year: 1984 – year: 2016 article-title: Complexity analysis of root clustering for a complex polynomial – start-page: 473 year: 1993 end-page: 478 – volume: 2 article-title: The Design and Analysis of Spatial Data Structures. publication-title: Addison Wesley, 1990. – year: 2006 – year: 1989 – year: 2000 – volume: 2014 start-page: 277 year: 2014 end-page: 282 – start-page: 201 year: 2000 end-page: 290 – volume: 17 start-page: 521 issue: 4 year: 1980 end-page: 529 article-title: A simple test for accuracy of approximate solutions to nonlinear (or linear) systems publication-title: SIAM J. Numer. Anal. – volume: 45 start-page: 511 issue: 2 year: 2013 end-page: 516 article-title: Voronoi Diagrams of algebraic distance fields publication-title: Computer Aided Design – volume: 8503 start-page: 38 year: 2014 end-page: 49 – volume: 2 start-page: 365 year: 1987 end-page: 394 article-title: An ( log ) algorithm for the Voronoi diagram for a set of simple curve segments publication-title: Discrete and Comp. Geom. – start-page: 760 end-page: 795 article-title: Adaptive isotopic approximation of nonsingular curves: the parameterizability and nonlocal isotopy approach – volume: 45 start-page: 760 issue: 4 year: 2011 end-page: 795 article-title: Adaptive isotopic approximation of nonsingular curves: the parameterizability and nonlocal isotopy approach publication-title: Discrete and Comp. 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Eurographics Symposium on Geometry Processing year: 2004 ident: e_1_2_7_20_2 – start-page: 434 volume-title: Lect. Notes in C.S year: 2013 ident: e_1_2_7_26_2 – ident: e_1_2_7_2_2 doi: 10.1016/B978-044482537-7/50006-1 – volume-title: Lecture Notes in Computer Science, No. 400 year: 1989 ident: e_1_2_7_11_2 |
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| SubjectTerms | Algorithms Anisotropy Categories and Subject Descriptors (according to ACM CCS) F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems-Geometrical problems and computations G.1.0 [Numerical Analysis]: General-Interval arithmetic G.1.2 [Numerical Analysis]: Approximation-Approximation of surfaces and contours G.4 [Mathematical Software]: -Algorithm design and analysis I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Geometric algorithms Image processing systems Mathematical analysis Mathematical models Minimization Optimization Scalars Studies Subdivisions |
| Title | Planar Minimization Diagrams via Subdivision with Applications to Anisotropic Voronoi Diagrams |
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