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15
Topological Relations Between Regions With Holes
 Int. Journal of Geographical Information Systems
, 1994
"... The 4intersection, a model for the representation of topological relations between 2dimensional objects with connected boundaries and connected interiors, is extended to cover topological relations between 2dimensional objects with arbitrary holes, called regions with holes. Each region with hole ..."
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Cited by 88 (6 self)
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The 4intersection, a model for the representation of topological relations between 2dimensional objects with connected boundaries and connected interiors, is extended to cover topological relations between 2dimensional objects with arbitrary holes, called regions with holes. Each region with holes is represented by its generalized regionthe union of the object and its holes and the closure of each hole. The topological relation between two regions with holes, A and B, is described by the set of all individual topological relations between (1) A 's generalized region and B's generalized region, (2) A 's generalized region and each of B's holes, (3) B's generalized region with each of A 's holes, and (4) each of A 's holes with each of B's holes. As a side product, the same formalism applies to the description of topological relations between 1spheres. An algorithm is developed that minimizes the number of individual topological relations necessary to describe a configuration completely. This model of representing complex topological relations is suitable for a multilevel treatment of topological relations, at the least detailed level of which the relation between the generalized regions prevails. It is shown how this model applies to the assessment of consistency in multiple representations when, at a coarser level of less detail, regions are generalized by dropping holes.
Evaluating Inconsistencies among Multiple Representations
 Sixth International Symposium on Spatial Data Handling
, 1994
"... If Geographic Information Systems (GISs) contain multiple representations of the same geographic objects at different levels of detail, it becomes necessary to compare the different representations and assess whether they contradict each other or not. Topological information is generally considered ..."
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Cited by 42 (5 self)
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If Geographic Information Systems (GISs) contain multiple representations of the same geographic objects at different levels of detail, it becomes necessary to compare the different representations and assess whether they contradict each other or not. Topological information is generally considered firstclass geographic information and as such the preservation of topological relations among objects in different representations manifests a critical criterion for the comparison of multiple representations and their consistency evaluation. This paper describes a framework within which the topological consistency of multiple representations can be assessed. The rational for assessing topological similarity is the monotonicity assumption of a generalization, under which the topology of any object and any topological relation between objects must stay the same through consecutive representation levels; or continuously decrease in complexity and detail. Such changes are assessed through object similarity and relation similarity, respectively. Within this framework, only those topological invariants can be changed that are at least on an ordinal scale.
Towards a Formal Model for Multiresolution Spatial Maps
, 1995
"... . Topological and metric aspects of the multiresolution representation of geographic maps are considered. The combinatorial structure of maps is mathematically modelled through abstract cell complexes, and maps at different detail are related through continuous functions over such complexes. Metric ..."
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Cited by 38 (1 self)
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. Topological and metric aspects of the multiresolution representation of geographic maps are considered. The combinatorial structure of maps is mathematically modelled through abstract cell complexes, and maps at different detail are related through continuous functions over such complexes. Metric aspects of multiresolution are controlled through the concept of homotopy. Two alternative multiresolution models are proposed, which are implicitly defined by a sequence of map simplifications that fulfil both topological and metric consistency rules. 1 Introduction The representation of spatial data at different resolution in the context of a unified model is a topic of relevant interest in spatial information theory. Indeed, multiresolution modelling offers interesting capabilities for spatial representation and reasoning: from support to map generalisation and automated cartography [15], to efficient browsing over large GISs, to structured solutions in wayfinding and planning [25]. Curr...
A Proof System for Contact Relation Algebras
"... Contact relations have been studied in the context of qualitative geometry and physics since the early 1920s, and have recently received attention in qualitative spatial reasoning. In this paper, we present a sound and complete proof system in the style of Rasiowa & Sikorski (1963) for relatio ..."
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Cited by 17 (12 self)
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Contact relations have been studied in the context of qualitative geometry and physics since the early 1920s, and have recently received attention in qualitative spatial reasoning. In this paper, we present a sound and complete proof system in the style of Rasiowa & Sikorski (1963) for relation algebras generated by a contact relation. 1 Introduction Contact relations arise in the context of qualitative geometry and spatial reasoning, going back to the work of de Laguna (1922), Nicod (1924), Whitehead (1929), and, more recently, of Clarke (1981), Cohn et al. (1997), Pratt & Schoop (1998, 1999) and others. They are a generalisation of the "overlap relation" , obtained from a "part of" relation, which for the first time was formalised by Lesniewski (1916), (see also Lesniewski, 1983). One of Lesniewski's main concerns was to build a paradoxfree foundation of Mathematics, one pillar of which was mereology 1 or, as it was originally called, the general theory of manifolds or colle...
A Necessary Relation Algebra for Mereotopology
 Studia Logica
, 2001
"... We show that the basic operations of the relational calculus on a "contact relation" generate at least 25 relations in any model of the Region Connection Calculus [33], and we show how to interpret these relations in the collection of regular open sets in the twodimensional Euclidean plan ..."
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Cited by 13 (4 self)
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We show that the basic operations of the relational calculus on a "contact relation" generate at least 25 relations in any model of the Region Connection Calculus [33], and we show how to interpret these relations in the collection of regular open sets in the twodimensional Euclidean plane. 1 Introduction Mereotopology is an area of qualitative spatial reasoning (QSR) which aims to develop formalisms for reasoning about spatial entities [1, 12, 30, 31]. The structures used in mereotopology consist of three parts: 1. A relational (or mereological) part, 2. An algebraic part, 3. A topological part. The algebraic part is often an atomless Boolean algebra, or, more generally, an orthocomplemented lattice, both without smallest element. Due to the presence of the binary relations "partof" and "contact" in the relational part of mereotopology, composition based reasoning with binary relations has been of interest to the QSR community, and the expressive power, consistency and complexity o...
Relation Algebras over Containers and Surfaces: An Ontological Study of a Room Space
 SPATIAL COGNITION AND COMPUTATION
, 1999
"... Recent research in geographic information systems has been concerned with the construction of algebras to make inferences about spatial relations by embedding spatial relations within a space in which decisions about compositions are derived geometrically. We pursue an alternative approach by studyi ..."
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Cited by 10 (0 self)
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Recent research in geographic information systems has been concerned with the construction of algebras to make inferences about spatial relations by embedding spatial relations within a space in which decisions about compositions are derived geometrically. We pursue an alternative approach by studying spatial relations and their inferences in a concrete spatial scenario, a room space that contains such manipulable objects as a box, a ball, a table, a sheet of paper, and a pen. We derive from the observed spatial properties an algebra related to the fundamental spatial concepts of containers and surfaces and show that this containersurface algebra holds all properties of Tarski's relation algebra, except for the associativity. The crispness of the compositions can be refined by considering the relative size of the objects) and their roles (i.e., whether it is explicitly known that the objects are containers or surfaces).
A Qualitative Spatial Reasoner
 In the Proceedings of the 6th International Symposium on Spatial Data Handling (SDH
, 1994
"... Traditionally, GISs employ purely quantitative methods to represent and infer spatial information. This approach has serious shortcomings when dealing with qualitative spatial information, which may be incomplete or imprecise and without knowledge of the particular geometry of the spatial objects in ..."
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Cited by 9 (3 self)
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Traditionally, GISs employ purely quantitative methods to represent and infer spatial information. This approach has serious shortcomings when dealing with qualitative spatial information, which may be incomplete or imprecise and without knowledge of the particular geometry of the spatial objects involved. This paper describes efforts to build a prototype of a qualitative spatial reasoner about spatial relations such as topological relations, cardinal directions, and approximate distances. It builds on relation algebras developed for the individual spatial relations. The system is extensible as demonstrated by the inclusion of temporal relations. The novel concept in this objectoriented setting is the treatment of relations as firstclass objects, rather than as labeled links between spatial objects.
Map Cube Model  a model for multiscale data
 in Proceedings of Eighth International Symposium on Spatial Data Handling
, 1998
"... In this paper, we develop a model for the structure of multiscale data derived from a map series. First we present a model of a map. We give an ontology of map elements and assign them to the objects of the model.. Then we derive a model for a series of maps, connecting several of the map models wi ..."
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Cited by 8 (1 self)
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In this paper, we develop a model for the structure of multiscale data derived from a map series. First we present a model of a map. We give an ontology of map elements and assign them to the objects of the model.. Then we derive a model for a series of maps, connecting several of the map models with the help of tree structures. This model is called the map cube model, the horizontal axes denoting 2D space and the vertical axis denoting level of detail. A formal model of the map model and of the map cube model is given in Gofer.
Inconsistency Issues in Spatial Databases
 Lecture Notes In Computer Science, number 3300
, 2005
"... Abstract. This chapter analyzes inconsistency issues in spatial databases. In particular, it reviews types of inconsistency, specification of integrity constraints, and treatment of inconsistency in multiple representations and data integration. The chapter focuses on inconsistency associated with t ..."
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Cited by 6 (1 self)
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Abstract. This chapter analyzes inconsistency issues in spatial databases. In particular, it reviews types of inconsistency, specification of integrity constraints, and treatment of inconsistency in multiple representations and data integration. The chapter focuses on inconsistency associated with the geometric representation of objects, spatial relations between objects, and composite objects by aggregation. The main contribution of this paper is a survey of existing approaches to dealing with inconsistency issues in spatial databases that emphasizes the current state of the art and that outlines research issues in the context of inconsistency tolerance. 1
Volumes From Overlaying 3D Triangulations in Parallel
 Advances in Spatial Databases: Third Intl. Symp., SSD’93, volume 692 of Lecture Notes in Computer Science
, 1993
"... Consider a polyhedron that is triangulated into tetrahedra in two different ways. This paper presents an algorithm, and hints for implementation, for finding the volumes of the intersections of all overlapping pairs of tetrahedra. The algorithm should parallelize easily, based on our experience w ..."
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Cited by 4 (3 self)
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Consider a polyhedron that is triangulated into tetrahedra in two different ways. This paper presents an algorithm, and hints for implementation, for finding the volumes of the intersections of all overlapping pairs of tetrahedra. The algorithm should parallelize easily, based on our experience with similar algorithms. One application for this is, when given data in terms of one triangulation, to approximate it in terms of the other triangulation. One part of this algorithm is useful by itself. That is to locate a large number of points in a triangulation, by finding which tetrahedron contains each point. Keywords: tetrahedron, triangulation, overlay, uniform grid, finite element model, mass property, uniform grid, parallel, point location Contents 1 Introduction 2 2 Mathematical Foundation 3 # Until Dec 30, 1992: c/o Prof. Leila De Floriani, Dipartimento di Informatica e Scienze dell'Informazione, Universit a di Genova, Viale Benedetto XV, 3, 16132 GENOVA, ITALY, Phone: +3...