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Voronoi Diagram Presenter: GI1 11號 蔡逸凡 Reference: Computational Geometry ch7 ISBN: 3-540-65620-0 Outline Introduction Definitions Incremental Algorithm Fortune’s Algorithm Hardware Algorithm Applications 7-11 Shoppers … Where do people go to shop? Which location is suitable for new store? Assumptions price is the same at every site cost = price + transportation transportation = (Euclidean distance) (price per unit distance) consumers try to minimize the cost. Figure 7.1 The trading areas of the capitals of the twelve provinces in the Netherlands, as predicted by the Voronoi assignment model Introduction (cont) Fire observation towers Towers on fire Facility location: center of largest empty circle Path planning: stay as far away from all obstacles as possible Definitions Vor(P) [Voronoi diagram of P]: subdivision of the plane into n cells, one for each site in P, P p1 , p2 ,..., pn [Voronoi cell of pi]:the cell of Vor(P) that corresponds to a site pi V ( pi ) {x : x pi x p j , j i} V(P): the set of all points that have more than one nearest neighbor Definition (Cont) bisector h (p, q) p3 r hq, p distr , q distr , p p2 p1 h (q, p) Two sites Circumcircle of p1p2p3 Three sites Observation 7.1 V pi 1 j n , j i h pi , p j Theorem 7.2 Let P be a set of n point sites in the plane. If all the sites are collinear then Vor(P) consists of n-1 parallel lines. Otherwise, Vor(P) is connected and its edges are either segments or half-lines. Theorem 7.3 For n≧3, the number of vertices in the Voronoi diagram of a set of n point sites in the plane is at most 2n-5 and the number of edges is at most 3n-6. Give a bound to number of Voronoi edges and vertices CP(q): largest empty circle of q w.r.t. P To characterize which bisectors and intersections define features of the Voronoi diagram we make the following: not contain any site of P Theorem 7.4 For the Voronoi diagram Vor(P) of a set of points P the following holds: (i) A point q is a vertex of Vor(P) if and only if its largest CP(q) empty circle contains three or more sites on its boundary. pi q pj pk Theorem 7.4 (Cont) (ii) The bisector between sites pi and pj defines an edge of Vor(P) if and only if there is a point q on the bisector such that CP(q) contains both pi and pj on its boundary but no other site. pi q pj Theorem 7.4 (Cont) (ii) The bisector between sites pi and pj defines an edge of Vor(P) if and only if there is a point q on the bisector such that CP(q) contains both pi and pj on its boundary but no other site. pk pi q pj distq, pi distq, pi distq, pk for all 1 k n Incremental Algorithm Suppose that we have already built the Voronoi diagram Vp-1, and would like to add a new sites sp. First, find the site, say si, whose Voronoi polygon contains sp, and draw the perpendicular bisector between sp and si, denoted by B(sp, si). The bisector crosses the boundary of V(si) at two points, point x1 and point x2. Site sp is to the left of the directed line segment x1x2. The line segment x1x2 divides the Voronoi polygon V(si) into two pieces, the one on the left belonging to the Voronoi polygon of sp. Thus, we get a Voronoi edge on the boundary of the Voronoi polygon of si. Incremental Algorithm-2 Starting with the edge x1x2, expand the boundary of the Voronoi polygon of sl by the following procedure. The B(si, sl) crosses the boundary of V(si) at x2, entering the adjacent Voronoi polygon, say V(sj). Therefore, next draw the B(si, sj), and find the point, x3, at which the bisector crosses the boundary of V(sj). Similarly, find the sequence of segments of perpendicular bisectors of s and the neighboring sites until we reach the starting point x1. Let this sequence be (x1x2, x2x3, …, xm-1xm, xmx1). This sequence forms a CCW boundary of the Voronoi polygon of the new site s. Finally, we delete from Vp-1 the substructure inside the new Voronoi polygon, and thus get Vp. Example Example Example Example Example O(n) per point insertion Total complexity: O(n2) Complexity The method describe before: per Voronoi cell: On log n total: On2 log n plane sweep algorithm— Fortune’s algorithm: On log n Fortune’s Algorithm (1985) Use plane sweep to explore the Voronoi structure Sweep line The loci of equal distance points between the focus and the directrix is a parabola Fortune (cont) As the sweep line moves, more parabolas got generated The intersection between parabola is the trace of equal distance point between the two sites Beach line: monotone union of parabolic arcs Break point: where two arcs meet Site Event (Voronoi Edge) The branch starts when the sweep line first touches the site, forming a degenerate parabola (a line) Circle Event (Voronoi Vertex) The Voronoi edge comes to an end when the circumcircle pass the sweep line At such points, the corresponding arc got removed from the beach line Plane Sweep Algorithm (Cont) l :sweep line l :The closed half-plane above l q l 12 q s1 s3 s2 q Plane Sweep Algorithm (Cont) beach line breakpoint Observation 7.5 The beach line is x-monotone, that is, every vertical line intersects it in exactly one point. p1 p4 p3 p2 Process – Site event When reaching a new site, we consider the events where a new arc appears on the beach line Process – Site event (Cont) What happen to the Voronoi diagram at a site event? Lemma 7.6 The only way in which a new arc can appear on the beach line is through a site event. Process – Circle event Lemma 7.7 The only way in which an existing arc can disappear from the beach line is through a circle event. The detection of circle event Does not define a potential circle event False alarm This circle event helps to terminate the Voronoi edges Sweep line intersects the circumcircle This, however, does not generate circle event: cir.circle does not intersect sweep line [site coincide w/ circle event] Summary Site event: encounter a new site - get new arc Circle event: A new triple has converging breakpoint – has a circle event insert into the event queue The new arc is in the middle – never cause Check disappear triples that have circle event – false alarm, delete it from event queue Structure Event queue: the priority of an event is its y-coordinate Site event: store the site Circle event: lowest point of the circle Doubly-connected edge list Binary search tree T Structure (Cont) Binary search tree T breakpoint Site that define the arc Example 1 p3 p1 p2 Q={p1,p2,p3} Q={p2,p3} D={} D={} T: p1 Example 1 (Cont) Q={p3} D={e1} T: <p2,p1>->e1 p2 p1 Example 1 (Cont) Q={} T: <p1,p3>->e2 D={e1,e2} e1<-<p2,p1> p2 p1 p3 Example 1 (Cont) Q={c1} <p1,p3>->e2 T: D={e1,e2} e1<-<p2,p1> p2 p1 c1 p3 Example 1 (Cont) Q={} T: D={e1,e2,e3} e3<-<p2,p3> p2 p3 Example 1 (Cont) Q={} T: D={e1,e2,e3} e3<-<p2,p3> p2 A visual implementation of Fortune's Voronoi algorithm p3 Example 2 Q={p1,p2,p3,p4} Q={p2,p3,p4} D={} D={} T: p1 Example 2 (Cont) Q={p3,p4} T: <p1,p2>->e1 D={e1} p1 p2 Example 2 (Cont) Q={p4} T: <p1,p2>->e1 D={e1,e2} e2<-<p3,p1> p3 p1 p2 Q={c1,p4} T: <p1,p2>->e1 D={e1,e2} e2<-<p3,p1> p3 p1 c1 p2 Example 2 (Cont) Q={p4} T: D={e1,e2,e3} e3<-<p3,p2> p3 p2 Q={} T: <p4,p2>->e5 D={e1,e2,e3 e4,e5} e4<-<p3,p4> p3 p4 p2 Example 2 (Cont) Q={} T: <p4,p2>->e5 D={e1,e2,e3 e4,e5} e4<-<p3,p4> p3 p4 p2 Example 3 Q={p1,p2,p3,p4} Q={p2,p3,p4} D={} D={} T: p1 Example 3 (Cont) Q={p3,p4} T: <p2,p1>->e1 D={e1} p2 p1 Example 3 (Cont) Q={p4} T: <p1,p3>->e2 D={e1,e2} e1<-<p2,p1> p2 p1 p3 Q={p4,c1} T: <p1,p3>->e2 D={e1,e2} e1<-<p2,p1> p2 p1 c1 p3 Example 3 (Cont) Example 3 (Cont) Q={p4} <p1,p3>->e2 T: D={e1,e2} e1<-<p2,p1> p2 p1 p3 Example 3 (Cont) Example 3 (Cont) Proofs Theorem 7.2 (Cont) Proof: (1) Collinear: Vor(P) consists of n-1 parallel lines ．．． 1 2 3 n-1 n n – 1 parallel lines Theorem 7.2 (Cont) Proof: (2) Vor(P) is connected and its edges are either segments or half-lines. → suppose : an edge e of Vor(P) that is a full line f Theorem 7.2 (Cont) Proof: (2) Vor(P) is connected and its edges are either segments or half-lines. → suppose : an edge e of Vor(P) that is a full line Theorem 7.3 (Cont) Euller' s formula : mv me m f 2 nv 1 ne n 2 2ne 3nv 1 (7.1) (7.2) from (7.2)得 : 2 ( 7.1) ne 1 代入 ne 3n 6 3 3 3 代入 ( 7.1) ne nv nv 2n 5 2 2 nv Lemma 7.6 (Cont) Proof: First possibility: suppose an already existing parabola j break through the beach line, is defined by a site p j Lemma 7.6 (Cont) Proof: Let l y denote the y-coordinate of the sweep line p j p j,x , p j, y 12 The parabola j is given by: j y 1 2 p j , y x l y 2 2 p j , x x p 2j , x p 2j , y l y2 Lemma 7.6 (Cont) Proof: Second possibility: j appear in between two arcs i and k Lemma 7.6 (Cont) Proof: Lemma 7.6 (Cont) Consequence: The beach line consists of at most 2n-1 parabolic arcs.