Dr. Marina Gavrilova

1. Computational Geometry Algorithms: Voronoi Diagram and Delaunay Triangulation Dr. Marina Gavrilova Associate Professor,Department of Computer Science, University of Calgary, Calgary, Alberta, Canada.

2. Lecture Overview • General Research Areas • Voronoi diagram introductions • Voronoi diagram theory • Application examples • Conclusions

3. Computational Geometry in Action

4. Activities in Brief Founder and Co-Director: • Biometric Technologies Laboratory, CFI • SPARCS Laboratory for Spatial Analysis in Computational Sciences, GEOIDE Editor-in-Chief: • International Journal LNCS Transactions on Computational Sciences, Springer-Verlag Guest Editor: • Int. Journal of Computational Geometry and Applications • IEEE Robotics and Automation Magazine RAM

5. Activities in Brief Chair and Co-Founder: • International Conference on Computational Sciences and Applications since 2003 • International Workshop on Computational Geometry and Applications (since 2001) • Chair, the 3rd International Symposium on Voronoi Diagrams and Applications 06 Author of new books:S. Yanushkevich, M. Gavrilova, P. Wang and S. Srihari "Image Pattern Recognition: Synthesis and Analysis in Biometrics," Series in Machine Perception and Artificial Intelligence, World Scientific Bestseller, 2007 • M. Gavrilova “Computational Intelligence: A Geometry-Based Approach,” Series on Studies in Computational Intelligence, Springer-Verlag, 2008

6. Areas of Research • Topological properties of data sets • Terrain rendering and surface triangulation • Path planning and obstacle avoidance • Robotics and Navigation • Autocorrelation analysis • Spatial-temporal models • Nearest neighbor properties • Terrain reconstruction and triangulation • Biological systems modeling • Molecular systems representation and analysis • Geographical Information Systems • Biometric analysis and synthesis • Granular-type materials simulation

7. Areas of Research Coral models (with J. Kaandorp) Dynamic data structures (with I. Kolingerova) Lipid bi-layers and molecular modeling (with N.N. Medvedev) Biometric research Porous materials Terrain modeling

8. Voronoi Diagrams – A Brief Overview

9. Voronoi diagrams in selected applications (ISVD 2006) A. Mukhopadhyay, S. Das, Canada L. Wang et. al. China M. Moriguchu and K. Sugihara, Japan Tetsuo Asano, Japan James Dean Palmer, USA T. Taylor and I. Vaisman, USA Deok-Soo Kim, Korea C. Gold and M. Dakowicz, UK P. Bhattacharya and M. Gavrilova, Canada

10. Voronoi diagrams in tiling (ISVD 2006) Craig S. Kaplan, University of Waterloo, Canada Jos Leys, Belgium

11. Voronoi diagram and Delaunay Tessellation Voronoi diagram is one of the fundamental computational geometry data structures that stored proximity information for a set of obejcts. It’s dual structure, often used in computer graphics, is Delaunay Tessellation. A generalized Voronoi diagram(GVD) for a set of objects in space is the set of generalized Voronoi regions where d(x,P)is a distance function between a point x and a site P in the d-dimensional space.

12. Delaunay Tessellation Ageneralized Delaunay tessellation (triangulation in 2d) is the dual of the generalized Voronoi diagram obtained by joining all pairs of sites whose Voronoi regions share a common Voronoi edge according to some specific rule.

13. Delaunay Triangle and Void Empty volume Delaunay simplex (tetrahedron in 3D) defines a simplicial configuration of spheres and a void (empty space) between spheres.

14. Voronoi diagram and Delaunay triangulation in 2D

15. Main properties of the Voronoi Diagram Under assumptions that no four sites from the object (generator) set S are cocircular: • Voronoi vertex is the intersection of 3 Voronoi edges and a common point of 3 Voronoi regions • Voronoi vertex is equidistant from 3 sites. It lies in the center of a circle inscribed between 3 cites • Empty circle property This inscribed circle is empty, i.e. it does not contain any other sites • Nearest-neighbor property If Q is the nearest neighbor of P then their Voronoi regions share an edge (to find a nearest neighbor it is sufficient to check only neighbors in the VD)

16. Main properties of the Delaunay Triangulation Under assumptions that no three sites from the set S (generator) lie on the same straight line: • The straight-line dual of the Voronoi diagram is a triangulation of S • The circumcircle of any Delaunay triangle does not contain any points of S in its interior • If each triangle of a triangulation of the convex hull of S satisfies the empty circle property, then this triangulation is the Delaunay triangulation of S . • If Q is the nearest neighbor of P then their Voronoi regions share an edge (to find a nearest neighbor it is sufficient to check only neighbors in the VD).

17. Distance metrics for Voronoi Diagrams General Lp distance Manhattan Supremum Manalanobis

18. Voronoi diagrams • VD: Thiessen polygons, Delaunay triangulations, tessellations

19. Tessellation • Tessellation is often obtained using Delaunay triangulation.

20. Voronoi diagram • Given a set of N sites (points) in the plane or a 3D space • Distance function d(x,P) between point x and site P is defined according to some metric • Voronoi region Vor(P) is the set of all points which are closer to P than to any other site • Voronoi diagram is the union of all Voronoi regions

21. Voronoi Diagram

22. Voronoi diagram

23. Voronoi diagram

24. Voronoi diagram properties Assumption 1. No four points from the set S are cocircular. Property 1. Voronoi vertex is the intersection of 3 Voronoi edges and a common point of 3 Voronoi regions Property 2. Voronoi vertex is equidistant from 3 sites. It lies in the center of a circle inscribed between 3 cites Property 3. Empty circle property This inscribed circle is empty, i.e. it does not contain any other sites Property 4. Nearest-neighbor property If Q is the nearest neighbor of P then their Voronoi regions share an edge (to find a nearest neighbor it is sufficient to check only neighbors in the VD) Property 5. Voronoi region Vor(P) is unbounded if and only if P belongs to the boundary of convex hull of S.

25. VD properties

26. VD properties

27. VD properties

28. VD properties

29. VD properties

30. Delaunay triangulation Definition 3. A Delaunay triangulation (DT) is the straight-line dual of the Voronoi diagram obtained by joining all pairs of sites whose Voronoi regions share a common Voronoi edge [Delaunay 34]. Follows from the definition: • If two Voronoi regions Vor(P) and Vor(Q) share an edge, then sites P and Q are connected by an edge in the Delaunay triangulation • If a Voronoi vertex belongs to Vor(P), Vor(Q) and Vor(R), then DT contains a triangle (P,Q,R)

31. Delaunay triangulation

32. DT properties Assumption 2. No three points from the set S lie on the same straight line. Theorem. The straight-line dual of the Voronoi diagram is a triangulation of S [Preparata and Shamos 85]. Property 5. The circumcircle of any Delaunay triangle does not contain any points of S in its interior [Lawson 77]. Property 6. If each triangle of a triangulation of the convex hull of S satisfies the empty circle property, then this triangulation is the Delaunay triangulation of [Lawson 77].

33. DT properties The edge of the quadrilateral satisfies the localmin-max criterion if the following equation holds: A triangulation satisfies the global min-max criterion if every internal edge of a convex quadrilateral in the triangulation satisfies the local min-max criterion. Property 8. The Delaunay triangulation satisfies the global min-max criterion [Lawson 77]. Property 9. If a triangulation of the convex hull of satisfies the global min-max criterion then it is the Delaunay triangulation of [Lawson 77].

34. VD and DT

35. VD and DT • Both DT and VD effectively represent the proximity information for the set of sites. They can be easily transformed into each other. • VD contains geometrical information, while DT contains topological information.

36. Generalized Voronoi diagram • Given a set S of n sites (spheres) in d-dimensional space • Distance functiond(x,P) between a point x and a site P is defined.

37. Generalized Voronoi diagram A generalized Voronoi diagram(GVD) for a set of objects in space is the set of generalized Voronoi regions where d(x,P)is a distance function between a point x and a site P in the d-dimensional space.

38. Generalized Delaunay tessellation Ageneralized Delaunay triangulation (GDT) is the dual of the generalized Voronoi diagram obtained by joining all pairs of sites whose Voronoi regions share a common Voronoi edge.

39. Generalized distance functions Power Additively weighted Euclidean Manhattan supremum x P x d(x,P) P x r r p p P ( ) d x , P ( ) d x , P x P General metrics

40. P Q Q P B(P,Q) B(P,Q) Power and Euclidean Voronoi diagrams Power bisector Euclidean bisector Power diagram and Delaunay triangulation Euclidean diagram and Delaunay triangulation

41. Manhattan and Supremum VD Manhattan bisectors Supremum bisectors Manhattan diagram and Delaunay triangulation Supremum diagram and Delaunay triangulation

42. Properties of Generalized VD and DT • The vertex of generalized Voronoi diagram is a center of a sphere inscribed between d +1Voronoi sites (spheres). • The inscribed sphere is empty, i.e. it does not contain any other sites. • One of the facets of the generalized Voronoi region Vor(P) defines a nearest-neighbor of P. • A Voronoi region Vor(P) is unlimited if and only if site P belongs to the convex hull of S. • Thesphere inscribed between the sites comprising a simplex of generalized Delaunay tessellation is an empty sphere. • The power Delaunay tessellation of a set of spheres S is a tetrahedrization.

43. Algorithmic Strategies • Incremental • Divide-and-conquer • Sweep-line/plane • Dimension reduction • Geometric decomposition

44. Part 2 • Algorithm development techniques on example of Voronoi Diagrams • Incremental construction • Divide and conquer • Sweep-plane • Geometric transformations • Dynamic data structures

45. Incremental construction INCIRCLE condition • power • Polynomial of 4th order in the plane • Euclidean • Polynomial of 8th order in the plane • System of linear equations and inequalities in the plane • Supremum

46. Incremental construction • Incremental method outline • Insert new site P • Perform swap operations on quadrilaterals where the empty-sphere condition is not satisfied Method complexity isO(n2)in the plane.

47. Sweep-plane algorithm • Algorithm description • Throw pebbles in the water • Intersection of waves gives the Voronoi diagram • Add time as 3rd dimension: waves transform to pyramids. • Sweep pyramids with the sweep-plane to get the Voronoi diagram

48. Sweep-plane algorithm • Properties • The complexity of the sweep-plane algorithm in generalized Manhattan metric isO(n log n). • Sweep-plane method is not applicable to the power diagram construction.

49. Sweep-plane algorithm • An example of a sweep-plane construction of a Voronoi diagram in L1 metric

50. Sweep-plane algorithm Cones in L1