Results 1  10
of
189
Voronoi diagrams  a survey of a fundamental geometric data structure
 ACM COMPUTING SURVEYS
, 1991
"... This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. ..."
Abstract

Cited by 753 (5 self)
 Add to MetaCart
This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. The paper puts particular emphasis on the unified exposition of its mathematical and algorithmic properties. Finally, the paper provides the first comprehensive bibliography on Voronoi diagrams and related structures.
Geometric Shortest Paths and Network Optimization
 Handbook of Computational Geometry
, 1998
"... Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to ..."
Abstract

Cited by 194 (15 self)
 Add to MetaCart
(Show Context)
Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal
On Bregman Voronoi Diagrams
 in "Proc. 18th ACMSIAM Sympos. Discrete Algorithms
, 2007
"... The Voronoi diagram of a point set is a fundamental geometric structure that partitions the space into elementary regions of influence defining a discrete proximity graph and dually a wellshaped Delaunay triangulation. In this paper, we investigate a framework for defining and building the Voronoi ..."
Abstract

Cited by 60 (27 self)
 Add to MetaCart
(Show Context)
The Voronoi diagram of a point set is a fundamental geometric structure that partitions the space into elementary regions of influence defining a discrete proximity graph and dually a wellshaped Delaunay triangulation. In this paper, we investigate a framework for defining and building the Voronoi diagrams for a broad class of distortion measures called Bregman divergences, that includes not only the traditional (squared) Euclidean distance, but also various divergence measures based on entropic functions. As a byproduct, Bregman Voronoi diagrams allow one to define informationtheoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We show that for a given Bregman divergence, one can define several types of Voronoi diagrams related to each other
Straight Skeletons for General Polygonal Figures in the Plane
, 1996
"... : A novel type of skeleton for general polygonal figures, the straight skeleton S(G) of a planar straight line graph G, is introduced and discussed. Exact bounds on the size of S(G) are derived. The straight line structure of S(G) and its lower combinatorial complexity may make S(G) preferable to th ..."
Abstract

Cited by 44 (2 self)
 Add to MetaCart
(Show Context)
: A novel type of skeleton for general polygonal figures, the straight skeleton S(G) of a planar straight line graph G, is introduced and discussed. Exact bounds on the size of S(G) are derived. The straight line structure of S(G) and its lower combinatorial complexity may make S(G) preferable to the widely used Voronoi diagram (or medial axis) of G in several applications. We explain why S(G) has no Voronoi diagram based interpretation and why standard construction techniques fail to work. A simple O(n) space algorithm for constructing S(G) is proposed. The worstcase running time is O(n 3 log n), but the algorithm can be expected to be practically efficient, and it is easy to implement. We also show that the concept of S(G) is flexible enough to allow an individual weighting of the edges and vertices of G, without changes in the maximal size of S(G), or in the method of construction. Apart from offering an alternative to Voronoitype skeletons, these generalizations of S(G) have ap...
Surface reconstruction based on a dynamical system
 In Proc. 23rd Ann. Conf. European Association for Computer Graphics (Eurographics), Computer Graphics Forum
, 2002
"... We present an efficient algorithm that computes a manifold triangular mesh from a set of unorganized sample points in 3. The algorithm builds on the observation made by several researchers that the Gabriel graph of the sample points provides a good surface description. However, this surface descrip ..."
Abstract

Cited by 40 (3 self)
 Add to MetaCart
We present an efficient algorithm that computes a manifold triangular mesh from a set of unorganized sample points in 3. The algorithm builds on the observation made by several researchers that the Gabriel graph of the sample points provides a good surface description. However, this surface description is only onedimensional. We associate the edges of the Gabriel graph with index 1 critical points of a dynamical system induced by the sample points. Exploiting also the information contained in the critical points of index 2 provides a twodimensional surface description which can be easily turned into a manifold. 1.
The VisibilityVoronoi complex and its applications
 In Proc. 21st Annu. ACM Sympos. Comput. Geom. (SCG
, 2005
"... We introduce a new type of diagram called the VV (c)diagram (the Visibility–Voronoi diagram for clearance c), which is a hybrid between the visibility graph and the Voronoi diagram of polygons in the plane. It evolves from the visibility graph to the Voronoi diagram as the parameter c grows from 0 ..."
Abstract

Cited by 37 (4 self)
 Add to MetaCart
(Show Context)
We introduce a new type of diagram called the VV (c)diagram (the Visibility–Voronoi diagram for clearance c), which is a hybrid between the visibility graph and the Voronoi diagram of polygons in the plane. It evolves from the visibility graph to the Voronoi diagram as the parameter c grows from 0 to ∞. This diagram can be used for planning naturallooking paths for a robot translating amidst polygonal obstacles in the plane. A naturallooking path is short, smooth, and keeps — where possible — an amount of clearance c from the obstacles. The VV (c)diagram contains such paths. We also propose an algorithm that is capable of preprocessing a scene of configurationspace polygonal obstacles and constructs a data structure called the VVcomplex. The VVcomplex can be used to efficiently plan motion paths for any start and goal configuration and any clearance value c, without having to explicitly construct the VV (c)diagram for that cvalue. The preprocessing time is O(n 2 log n), where n is the total number of obstacle vertices, and the data structure can be queried directly for any cvalue by merely performing a Dijkstra search. We have implemented a Cgalbased software package for computing the VV (c)diagram in an exact manner for a given clearance value, and used it to plan naturallooking paths in various applications.
A generic framework for parcellation of the cortical surface into gyri using geodesic Voronoi diagrams
, 2003
"... In this paper, we propose a generic automatic approach for the parcellation of the cortical surface into labeled gyri. These gyri are defined from a set of pairs of sulci selected by the user. The selected sulci are first automatically identified in the data, then projected onto the cortical surface ..."
Abstract

Cited by 37 (5 self)
 Add to MetaCart
In this paper, we propose a generic automatic approach for the parcellation of the cortical surface into labeled gyri. These gyri are defined from a set of pairs of sulci selected by the user. The selected sulci are first automatically identified in the data, then projected onto the cortical surface. The parcellation stems from two nested Vorono diagrams computed geodesically to the cortical surface. The first diagram provides the zones of influence of the sulci. The boundary between the two zones of influence of each selected pair of sulci stands for a gyrus seed. A second diagram yields the gyrus parcellation. The distance underlying the Vorono diagram allows the method to interpolate the gyrus boundaries where the limiting sulci are interrupted. The method is illustrated with twelve different hemispheres.
Towards a LeagueIndependent Qualitative Soccer Theory for RoboCup
, 2005
"... The paper discusses a topdown approach to model soccer knowledge, as it can be found in soccer theory books. The goal is to model soccer strategies and tactics in a way that they are usable for multiple RoboCup soccer leagues, i.e. for different hardware platforms. We investigate if and how socc ..."
Abstract

Cited by 35 (12 self)
 Add to MetaCart
(Show Context)
The paper discusses a topdown approach to model soccer knowledge, as it can be found in soccer theory books. The goal is to model soccer strategies and tactics in a way that they are usable for multiple RoboCup soccer leagues, i.e. for different hardware platforms. We investigate if and how soccer theory can be formalized such that specification and execution is possible. The advantage is clear: theory abstracts from hardware and from specific situations in leagues.