Week 8 Voronoi Diagrams

Transcription

1 1 Week 8 Voronoi Diagrams

2 2 Voronoi Diagram Very important problem in Comp. Geo. Discussed back in 1850 by Dirichlet Published in a paper by Voronoi in 1908

3 3 Voronoi Diagram Fire observation towers: an example Given n fire observation towers Which tower must extinguish a starting fire? tower fire

4 4 Fire Observation Towers B tower A D fire C

5 5 Fire Observation Towers What if we start a separate fire at each tower? B A D C

6 6 Fire Observation Towers B A D C

7 7 Voronoi Diagram Nearest Neighbor Clustering Assume that we have 3 types of nuts Type A: Inner Diameter 1, Outer Diameter 2 Type B: Inner Diameter 3, Outer Diameter 4 Type C: Inner Diameter 2, Outer Diameter 4 A B C

8 8 Voronoi Diagram A certain quality assurance measurement device tests a bolt as having Inner Diameter 2.2 and Outer Diameter 3.8 Is it Type A, B or C? x

9 9 Voronoi Diagram Create a feature space inner radius 3 B 2 x 1 C A outer radius

10 10 Voronoi Diagram Check the Voronoi Polygons inner radius 3 B 2 x 1 C A outer radius

11 11 Voronoi Diagrams Suppose that you want to open a new store Where should you locate it for best revenues? City Store Store Store Store

12 12 Voronoi Diagrams You need to stay away from existing stores City Store Store Store? Store

13 13 Voronoi Diagrams Center of the largest empty circle! City Store Store Store? Store This center is always on the Voronoi Diagram!

14 14 Voronoi Diagrams Robot motion planning A robot walks through a field filled with obstacles Tries to avoid collisions What is the best path? If the obstacles are single points The best path lies on the conventional Voronoi Diagram

15 15 Voronoi Diagrams Crystallization

16 16 Formal Definition Let P={p1, p2,, pn} be a set of points in E2 These are called the sites Partition the plane by assigning every point to the nearest site. All points assigned to pi are the Voronoi region of pi V(pi) = {x : pi - x pj - x j i}

17 17 Formal Definition Some points do not have a unique nearest site, or nearest neighbor These form the Voronoi Diagram V(P) x x

18 18 Two Sites B12 bisector p1 p2

19 19 Three Sites B12 p2 circumcenter p1 B23 B13 p3

20 20 Halfplanes H p, p V p i = i j i j H3 Voronoi Region of pi p1 pi p2 H1 H2 p3

21 21 Voronoi Regions Note that each Voronoi Region is convex The edges are called the Voronoi Edges The vertices are called the Voronoi Vertices Non-degenerate: if three edges meet at a vertex Degenerate: if more than three edges meet at a vertex

22 22 Size of Diagram How many regions exist for n sites? Linear? How many edges exist for n sites? Quadratic? Not really!

23 23 Size of a Diagram Assume that all vertices are non-degenerate Construct the dual of the Voronoi Diagram The dual is a planar graph A planar graph of n vertices has O(n) edges O(n) faces This holds even with degenerate vertices

24 24 Delaunay Triangulations If the dual graph is drawn with straight lines we obtain a triangulation of the sites this is called the Delaunay Triangulation: D(P)

25 25 Delaunay Triangulation D(P) is the straight-line dual of V(P)

26 26 Delaunay Triangulation D(P) is a triangulation if no four points are cocircular

27 27 Delaunay Triangulation Each face of D(P) corresponds to a vertex of V(P) Each edge of D(P) corresponds to an edge of V(P) Each node of D(P) corresponds to a region of V(P)

28 28 Delaunay Triangulation The boundary of D(P) is the convex hull of sites

29 29 Delaunay Triangulation The interior of each face of D(P) contains no sites

30 30 Voronoi Diagrams Each Voronoi Region is convex

31 31 Voronoi Diagrams V(pi) is unbounded iff pi is on the convex hull

32 32 Voronoi Diagrams If v is a Voronoi vertex of regions V(p1), V(p2), V(p3), then it is the center of the circle C(v) through p1,p2,p3 p1 v p2 p3

33 33 Voronoi Diagrams C(v) is the circumcircle of the Delaunay Triangle of v p1 v p2 p3

34 34 Voronoi Diagrams The interior of C(v) contains no sites p1 v p2 p3

35 35 Voronoi Diagrams If pj is a nearest neighbor of pi, then (pi,pj) is an edge of D(P) p1 p2

36 36 Voronoi Diagrams If there is some circle through pi and pj that contains no other sites, then (pi,pj) is an edge of D(P). The reverse also holds: for every edge, there is an empty circle. p1 p2

37 37 Delaunay Triangulation First part: If ab is a Delaunay Edge, then there exists an empty circle through a and b. Voronoi Edge center a If we have this site inside the circle, then this site is closer to the center, hence the Voronoi Edge is not really a Voronoi Edge r r Delaunay Edge b

38 38 Delaunay Triangulation Second part: If there is an empty circle through a and b, then ab is a Delaunay Edge a b

39 39 Algorithms Intersection of Halfplanes Construct each Voronoi Region separately Intersection of n-1 halfplanes Can be computed in O(n log n) per region Overall O(n2 log n)

40 40 Algorithms Incremental Construction Suppose the Voronoi D. of k points is constructed Now, we add one more point Let the new point be in circles C(v1)...C(vm) Only these Voronoi points will be removed Changes are local Running time: O(n2)

41 41 Algorithms Divide and Conquer Asymptotically optimal But, difficult to implement Achieves O(n log n) time

42 42 Algorithms Fortune's Algorithm Plane sweep Construct the Voronoi Diagram of the swept region Major problem The edges of some Voronoi Regions are encountered before the corresponding sites! Fortune proposed a beautiful solution

43 43 The Problem Edges are encountered first

44 44 Cones intersection site site projection

45 45 Cone Slicing

46 46 Parabolic Front

47 47 Applications Nearest Neighbors Which is the nearest neighbor to a query point? How to do it? Construct the Voronoi Diagram in O(n log n) Find in which Voronoi Region the point falls we will later see that this can be done in O(log n)

48 48 Applications Fat triangulation of a point set Maximizing the minimum angle How to do it? Edelsbrunner proved that The Delaunay Triangulation is a fat triangulation

49 49 Applications Largest Empty Circle Remember the store location problem We are looking for a center inside the Convex Hull f(p): the radius of the largest empty circle at p f(p) cannot be maximum if the circle touches a single point

50 50 Largest Empty Circle we can move the center to obtain a larger radius

51 51 Largest Empty Circle How about touching two sites? we can move the center to obtain a larger radius

52 52 Largest Empty Circle How about touching three sites? now the circle cannot be moved without including one of the sites

53 53 Largest Empty Circle Therefore, the center of the largest empty circle must be a Voronoi Vertex or, the center is on the Convex Hull where the circle touches two sites An algorithm can be implemented in O(n log n)

54 54 Medial Axis The medial axis of a polygon P is the set of points inside P that have more than one closest point among the points of boundary of P.

55 55 Connection to Convex Hulls Consider the function z = x2+y2 Given a set of points in 2D, transform all points to 3D using the above function (x, y, x2 + y2)

56 56 Connection to Convex Hulls (xi,yi,x +y ) 2 i 2 i z y (xi,yi) x

57 57 Connection to Convex Hulls z y x

58 58 Connection to Convex Hulls z y x Delaunay Triangulation