F. Aurenhammer and H. Edelsbrunner. An optimal algorithm for constructing the weighted Voronoi diagram in the plane. Pattern Recogn., 17:251--257, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and the Voronoi diagram is of complexity O(m) it can be constructed within time O(m log m) Second, if the unit balls are Euclidean circles centered at p i , with possibly di#erent radii, then the Voronoi diagram of points p i with multiplicative weights results. Aurenhammer and Edelsbrunner [1] have shown that the bisector of two di#erently weighted points equals a circle. The Voronoi diagram contains O(n ) many edges, faces, and vertices; it can be computed in O(n ) time. See Aurenhammer and Klein [2] for more information on both types of Voronoi diagrams. There is a general ....

....B(p, q) This bisector point lies on the common boundary of R(p, q) and R(q,p) Proof. We define the function f : R R with f(#) d q (p #e) d p (p #e) 6) Apointp #e on the ray from p in direction e is on the bisector i# f(#) 0. We show that f is a convex function. For [0, 1] and # 1 ,# 2 [0, we have f(# 1 (1 )# 2 ) # q (p (# 1 (1 )# 2 )e # p ( # 1 (1 )# 2 )e) # q ( p # 1 e q) 1 ) p # 2 e (# 1 (1 )# 2 )# p (e) # q (p # 1 e q) 1 )# q (p # 2 e # p (# 1 e) # p (# 2 e) f(# 1 ) 1 )f(# 2 ) Furthermore, ....

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F. Aurenhammer and H. Edelsbrunner. An optimal algorithm for constructing the weighted Voronoi diagram in the plane. Pattern Recogn., 17:251--257, 1984.

....construct the Voronoi diagram and look for the optimal location either on a vertex of this diagram or on the boundary of the region R. Unfortunately, for weighted sites, the Voronoi diagram is known to have quadratic complexity in the worst case, and it can be constructed in optimal O(n ) time [2]. Thus, the optimal location, using the Voronoi diagram, can be found in O(n ) time [9] The first subquadratic algorithm for the weighted problem under L# metric and a rectangular R region was presented by Follert et al. 10] Their algorithm runs in O(n log n) time. In this paper we ....

F. Aurenhammer and H. Edelsbrunner "An optimal algorithm for for constructing the weighted Voronoi diagram in the plane", Pattern Recognition, 17(2), 1984, pp. 251--257.

....algorithms unique What are its properties Will the flip algorithm terminate or loop forever Here we describe an approach that puts anisotropic meshing on firm theoretical ground. In Section 3 we define anisotropic Voronoi diagrams, a generalization of multiplicatively weighted Voronoi diagrams [4]. Anisotropic Voronoi diagrams can be defined in any dimensionality. The geometric dual of an anisotropic Voronoi diagram is not generally a triangulation. We describe conditions in which the Voronoi cells are guaranteed to be entirely visible from their generating sites in Section 5. For the ....

....Every site in W is said to own Vor(W ) The anisotropic Voronoi diagram of V is the arrangement of the Voronoi cells Vor(W ) W V, W #= #, Vor(W ) #= # . Figure 3 depicts an example. Anisotropic Voronoi diagrams are a generalization of multiplicatively weighted Voronoi diagrams [4] in which the distance metric is anisotropic. If the metric tensor field M is isotropic (i.e. for some scalar field c, Mp = cpI for all p where I is the identity tensor) the anisotropic Voronoi diagram is the multiplicatively weighted Voronoi diagram. One odd characteristic that anisotropic ....

Franz Aurenhammer and Herbert Edelsbrunner. An Optimal Algorithm for Constructing the Weighted Voronoi Diagram. Pattern Recognition 17:251--257, 1984.

....pointed at that, when the set of reference points does not contain any pair of adjacent points, the watersheds of a Euclidean distance map of a reference set R induce the discrete Voronoi diagram of R. In the same way, the weighted distance maps relate to the discrete generalized Voronoi diagrams [2, 12]. Generalized distance maps are based on the definition of the distance to a weighted reference set. Definition 2 A weighted reference set of a domain D is a tuple (R; w) where R is a subset of D and w is a function from R to IN. The function w is called a weight function, and the weighted ....

F. Aurenhammer and H. Edelsbrunner. An optimal algorithm for constructing the weighted voronoi diagram in the plane. Pattern Recognition, 17:251--257, 1984.

....and the Voronoi diagram is of complexity O(m) it can be constructed within time O(m log m) Second, if the unit balls are Euclidean circles centered at p i , with possibly di#erent radii, then the Voronoi diagram of points p i with multiplicative weights results. Aurenhammer and Edelsbrunner [1] have shown that the bisector of two di#erently weighted points equals a circle. The Voronoi diagram contains O(n 2 ) many edges, faces, and vertices; it can be computed in O(n 2 ) time. See Aurenhammer and Klein [2] for more information on both types of Voronoi diagrams. There is a general ....

....R with f(#) dq (p #e) dp(p #e) 5) Apointp #e is on the bisector i# f(#) 0. q p u q p u q p u q p u q p u Figure 2: The ray # pumay contain zero, one, or two bisector points, or, in a special case, a bisector segment or ray. We show that f is a convex function. For # [0, 1] and #1,# 2 # [0, #)wehave f(#1 (1 )#2) #q (p (#1 (1 )#2)e q) #p( #1 (1 )#2 )e) #q ( p #1e q) 1 ) p #2e q) #p(#1e (1 )#2e) # #q (p #1e q) 1 )#q (p #2e q) #p(#1e) 1 )#p(#2e) f(#1) 1 )f(#2) Furthermore, f(0) dq (p) is positive ....

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F. Aurenhammer and H. Edelsbrunner. An optimal algorithm for constructing the weighted Voronoi diagram in the plane. Pattern Recogn., 17:251--257, 1984.

....following. Visibility graphs [9, 26, 27, 51, 66] use features of the C obstacles to build line of sight type roadmaps. Retraction Methods [17, 20, 57] use regions and features of the C obstacles to build retraction (greatest clearance) minimalistic roadmaps (most often Voronoi 13 diagrams [10]) The Silhouette Method [18] is the only known complete algorithm which is only singly exponential in the number of degrees of freedom of the robot. It is often cited as evidence that no computationally simpler algorithm is likely to exist for motion planning. By sweeping one dimension at a ....

F. Aurenhammer and H. Edelsbrunner, \An optimal algorithm for constructing the weighted Voronoi diagram in the plane," Pattern Recogn., vol. 17, pp. 251{ 257, 1984.

.... O(n log n) solution for the case that the bounding region is the convex hull of the input points, and an O(n 2 log n) solution for case (b) For weighted sites, the Voronoi diagram is known to have quadratic complexity in the worst case, and it can be constructed in optimal O(n 2 ) time [4]. The optimal location is either a vertex of this diagram, or it lies on the boundary of the region R. For simple bounding regions with constant complexity, an optimal location can thus be found in O(n 2 ) time [8] In this paper, we present the first subquadratic algorithms for this problem. ....

F. Aurenhammer, H. Edelsbrunner, "An optimal algorithm for constructing the weighted Voronoi diagram in the plane", Pattern Recognition, 17(2), 1984, pp. 251-257.

....OO2VD (OOKVD) edges, so they require no additional supporting structures. 5 Multiplicatively Weighted Voronoi Diagram (MWVD) So far it has been implicitly assumed that each generator point has the same weight. However, this assumption may not be appropriate in many settings (e.g. Boots, 1980; Aurenhammer and Edelsbrunner, 1984). Rather, we would like the ability to adjust weights reflecting the variable properties of the generator points for instance, to reflect the attractiveness of some object such as a shopping centre or a city. The basic properties of an MWVD are again described by Okabe et al. 1992) and ....

Aurenhammer, F. and Edelsbrunner, H. (1984). An optimal algorithm for constructing the weighted Voronoi diagram in the plane, Pattern Recognition, 17 (2): 251-257.

....the weights of 1 p are 3 p different. The intersection of a line and a circle must be computed in this case. Lastly, Figure 15(c) shows the case where all weights are different, and consequently all bisectors will take the form of circular arcs. 6.3. 2 The centre and radius of Apollonius circle Aurenhammer and Edelsbrunner (1984) give the formulae for deriving the centre and radius of Apollonius circle. The centre of Apollonius circle is defined as: 2 2 2 2 q w p w q p w p q w , whereas the Apollonius radius is: 2 2 q w p w q p d q w p w e (when, q ....

Aurenhammer, F. and Edelsbrunner, H. (1984). An optimal algorithm for constructing the weighted Voronoi diagram in the plane, Pattern Recognition, 17 (2): 251-257.

.... separation diagram its sites are the points and great circle arcs of the bad viewpoint arrangement; however its bisectors are more complicated, being defined by the equation Delta( d; A(a; b) Delta( d; A(a 0 ; b 0 ) This form is similar to the multiplicatively weighted Voronoi variant [1, 2], but with an extra sine factor. For the restricted case where we only consider vertex vertex occlusions, the bisectors are great circles and spherical ellipsoids. There is a trivial O(jV j 6 ) upper bound on the worstcase complexity of this diagram, and the Omega Gamma jV j 4 ) worst case ....

....case where we only consider vertex vertex occlusions, the bisectors are great circles and spherical ellipsoids. There is a trivial O(jV j 6 ) upper bound on the worstcase complexity of this diagram, and the Omega Gamma jV j 4 ) worst case lower bound example of Aurenhammer and Edelsbrunner [2] can be adapted to this variant. We conjecture that the lower bound is tight. The unrestricted case, which allows vertex edge edge vertex occlusions as Voronoi sites, is at least this complex. Open Problem 1 Develop a tight bound for the size of the observed separation diagram. The next ....

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F. Aurenhammer, H. Edelsbrunner: "An Optimal Algorithm for Constructing the Weighted Voronoi Diagram in the Plane" in Pattern Recognition, 1984; 17(2):251--257.

....(polygonal) convex distance function is defined by a convex polygon C with the origin in its interior. The distance from x to y is the real r0 so that the boundary 7 problem distance to x time citation additive weights w i e(s i ; x) O(n log n) multiplicative weights w i e(s i ; x) O(n 2 ) [6] Laguerre or power (e(s i ; x) 2 Gamma w i ) 1=2 O(n log n) 4] L p jjs i Gamma xjj p O(n log n) 22] convex distance function O(n log n) 10] abstract axiomatic O(n log n) 21] simple polygon geodesic O(n log 2 n) 3] crystal growth w i Delta SP (s i ; x) O(n 3 nS log S) 28] ....

F. Aurenhammer, H. Edelsbrunner, An optimal algorithm for constructing the weighted Voronoi diagram in the plane. Pattern Recognition 17:251--257, 1984.

....the Voronoi diagram and look for the optimal location either on a vertex of this diagram or on the boundary of the region R. Unfortunately, for weighted sites, the Voronoi diagram is known to have quadratic complexity in the worst case, and it can be constructed in optimal O(n 2 ) time [2]. Thus, the optimal location, using the Voronoi diagram, can be found in O(n 2 ) time [9] The first subquadratic algorithm for the weighted problem under L1 metric and a rectangular R region was presented by Follert et al. 10] Their algorithm runs in O(n log 4 n) time. In this paper we ....

F. Aurenhammer and H. Edelsbrunner "An optimal algorithm for for constructing the weighted Voronoi diagram in the plane", Pattern Recognition, 17(2), 1984, pp. 251--257.

....obtain a partition of the plane into regions around each site p i;j , each point in the region being closest (in the weighted) sense to its p i;j than any other. The MWVD for k(k Gamma 1) 2 points has combinatorial complexity Theta(k 4 ) and is computable in that time up to a constant factor [6]. The optimal location for the registration mark is then a point on some Voronoi edge, interior to the part, that maximizes the weighted distance to its site. This formulation thus eliminates a factor of log k from the 2nd term of the time complexity given previously. 4 An Example We used ....

F. Aurenhammer and H. Edelsbrunner. An optimal algorithm for constructing the weighted Voronoi diagram in the plane. Pattern Recognition, 17(2):251--257, 1984.

....pointed at that, when the set of reference points does not contain any pair of adjacent points, the watersheds of a Euclidean distance map of a reference set R induce the discrete Voronoi diagram of R. In the same way, the weighted distance maps relate to the discrete generalized Voronoi diagrams [2, 12]. Generalized distance maps are based on the definition of the distance to a weighted reference set. Definition 2 A weighted reference set of a domain D is a tuple (R; w) where R is a subset of D and w is a function from R to IN. The function w is called a weight function, and the weighted ....

F. Aurenhammer and H. Edelsbrunner. An optimal algorithm for constructing the weighted voronoi diagram in the plane. Pattern Recognition, 17:251--257, 1984.

.... separation diagram, its sites are the points and great circle arcs of the bad viewpoint arrangement; however its bisectors are more complex, being defined by the equation Delta( d; a; b) Delta( d; a 0 ; b 0 ) This form is similar to the multiplicatively weighted Voronoi variant [2,3], but with an extra sine factor. Consider the restricted case where E = Each vertex vertex occlusion (v i ; v j ) generates a point Voronoi site s a = d ij with a corresponding weight w a = ae(v i ; v j ) By symmetry of the bad viewpoint arrangement, each site s a has a corresponding site ....

....diagram ( E = on jSj point sites lies within the bounds Omega (jSj 2 ) and O(jSj 2 2 ff(jSj) where ff is the inverse Ackermann function [1] Proof. An example of a diagram with Omega (jSj 2 ) size is given in Fig. 7. It is a simple adaption of Aurenhammer and Edelsbrunner s [3] worst case example for the multiplicatively weighted Voronoi diagram of points in the plane. To establish the upper worst case bound, consider an incremental construction of the diagram in non descending order of weight. When the site s t is inserted into OSD t Gamma1 , the newly created region ....

F. Aurenhammer, H. Edelsbrunner: "An Optimal Algorithm for Constructing the Weighted Voronoi Diagram in the Plane" in Patt. Recog., 1984; 17(2):251--257

.... O(n log n) solution for the case that the bounding region is the convex hull of the input points, and an O(n 2 log n) solution for case (b) For weighted sites, the Voronoi diagram is known to have quadratic complexity in the worst case, and it can be constructed in optimal O(n 2 ) time [4]. The optimal location is either a vertex of this diagram, or it lies on the boundary of the region R. For simple bounding regions with constant complexity, an optimal location can thus be found in O(n 2 ) time [8] In this paper, we present the first subquadratic algorithms for this problem. In ....

F. Aurenhammer, H. Edelsbrunner, "An optimal algorithm for constructing the weighted Voronoi diagram in the plane", Pattern Recognition, 17(2), 1984, pp. 251-257.

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F. Aurenhammer and H. Edelsbrunner. An optimal algorithm for constructing the weighted Voronoi diagram in the plane. Pattern Recogn., 17:251--257, 1984.

....sites. Still, the two models above are not equivalent. This becomes apparent when noticing that Voronoi diagrams with disconnected regions have no interpretation in the wavefront model. An example is the Voronoi diagram for point sites with multiplicative weights, in Aurenhammer and Edelsbrunner [5]. Being more important in the present context, straight skeletons are known to admit no distance from site de nition, in general; see Aichholzer and Aurenhammer [3] We now give characterizing conditions for the equivalence of both models. Call two sites i and j in Id non piercing if, for all ....

F.Aurenhammer, H.Edelsbrunner, An optimal algorithm for constructing the weighted Voronoi diagram in the plane. Pattern Recognition 17 (1984), 251-257.

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F. Aurenhammer and H. Edelsbrunner. An optimal algorithm for constructing the weighted Voronoi diagram in the plane. Pattern Recogn., 17:251-257, 1984.

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F. Aurenhammer and H. Edelsbrunner. An optimal algorithm for constructing the weighted Voronoi diagram in the plane. Pattern Recogn., 17:251-257, 1984.

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F. Aurenhammer and H. Edelsbrunner. An optimal algorithm for constructing the weighted Voronoi diagram in the plane. Pattern Recogn., 17:251-257, 1984.

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F. Aurenhammer and H. Edelsbrunner. An optimal algorithm for constructing the weighted Voronoi diagram in the plane. Pattern Recogn., 17:251-257, 1984.

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Aurenhammer F., Edelsbrunner H.: An optimal algorithm for for constructing the weighted Voronoi diagram in the plane. Pattern Recognition 17 (2) (1984) 251--257

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