Topic
Combinatorial map
About: Combinatorial map is a research topic. Over the lifetime, 75 publications have been published within this topic receiving 2256 citations.
Papers published on a yearly basis
Papers
TL;DR: The main result is that ordered topological models are (roughly speaking) equivalent with respect to the class of objects which can be modelled (i.e. withrespect to dimension and orientability).
Abstract: In boundary representation, a geometric object is represented by the union of a ‘topological’ model, which describes the topology of the modelled object, and an ‘embedding’ model, which describes the embedding of the object, for instance in three-dimensional Euclidean space. In recent years, numerous topological models have been developed for boundary representation, and there have been important developments with respect to dimension and orientability. In the main, two types of topological models can be distinguished. ‘Incidence graphs’ are graphs or hypergraphs, where the nodes generally represent the cells of the modelled subdivision (vertex, edge, face, etc.), and the edges represent the adjacency and incidence relations between these cells. ‘Ordered’ models use a single type of basic element (more or less explicitly defined), on which ‘element functions’ act; the cells of the modelled subdivision are implicitly defined in this type of model. In this paper some of the most representative ordered topological models are compared using the concepts of the n- dimensional generalized map and the n- dimensional map. The main result is that ordered topological models are (roughly speaking) equivalent with respect to the class of objects which can be modelled (i.e. with respect to dimension and orientability).
335 citations
TL;DR: The research develops several new algorithms, including one for computing the local kernel of a region, and a compact formal description of the topology and connectivity of the indoor structure represented by a connected, embedded graph.
Abstract: This article proposes a comprehensive approach to computing a navigation graph for an indoor space. It focuses on a single floor, but the work is easily extensible to multi-level spaces. The approach proceeds by using a formal model, based on the combinatorial map but enhanced with geometric and semantic information. The process is almost fully automatic, taking as input the building plans providing the geometric structure of the floors and semantics of the building, such as functions of interior spaces, portals, etc. One of the novel aspects in this work was the use of combinatorial maps and their duals to provide a compact formal description of the topology and connectivity of the indoor structure represented by a connected, embedded graph. While making use of existing libraries for the more routine computational geometry involved, the research develops several new algorithms, including one for computing the local kernel of a region. The process is evaluated by means of a case study using part of a university building.
113 citations
TL;DR: A necessary and sufficient condition for the regularity of rank 3 combinatorial maps is given in terms of Coxeter groups, and the difficulty in classifying the regular maps on surfaces is revealed.
69 citations
19 Nov 2003
TL;DR: This study studies the definitions of removal and contraction operations in the generalized maps framework to enable it to unambiguously represent the topology of a well-known class of subdivisions of n-dimensional (discrete) spaces.
Abstract: Removal and contraction are basic operations for several methods conceived in order to handle irregular image pyramids, for multi-level image analysis for instance. Such methods are often based upon graph-like representations which do not maintain all topological information, even for 2-dimensional images. We study the definitions of removal and contraction operations in the generalized maps framework. These combinatorial structures enable us to unambiguously represent the topology of a well-known class of subdivisions of n-dimensional (discrete) spaces. The results of this study make a basis for a further work about irregular pyramids of n-dimensional images.
65 citations
TL;DR: This paper defines and analyzes a simple robot with local sensors that moves in an unknown polygonal environment that can execute wall-following motions and can traverse the interior of the environment only when following parallel to an edge.
Abstract: This paper defines and analyzes a simple robot with local sensors that moves in an unknown polygonal environment. The robot can execute wall-following motions and can traverse the interior of the environment only when following parallel to an edge. The robot has no global sensors that would allow precise mapping or localization. Special information spaces are introduced for this particular model. Using these, strategies are presented to solve several tasks: 1) counting vertices, 2) computing the path winding number, 3) learning a combinatorial map, which is called the cut ordering, that encodes partial geometric information, and 4) solving pursuit-evasion problems.
56 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2020 | 1 |
| 2019 | 5 |
| 2018 | 1 |
| 2016 | 2 |
| 2015 | 5 |
| 2014 | 8 |