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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 68 (3 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.
Topological Queries in Spatial Databases
 Journal of Computer and System Sciences
, 1996
"... We study topological queries over twodimensional spatial databases. First, we show that the topological properties of semialgebraic spatial regions can be completely specified using a classical finite structure, essentially the embedded planar graph of the region boundaries. This provides an invar ..."
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Cited by 45 (2 self)
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We study topological queries over twodimensional spatial databases. First, we show that the topological properties of semialgebraic spatial regions can be completely specified using a classical finite structure, essentially the embedded planar graph of the region boundaries. This provides an invariant characterizing semialgebraic regions up to homeomorphism. All topological queries on semialgebraic regions can be answered by queries on the invariant whose complexity is polynomially related to the original. Also, we show that for the purpose of answering topological queries, semialgebraic regions can always be represented simply as polygonal regions. We then study query languages for topological properties of twodimensional spatial databases, starting from the topological relationships between pairs of planar regions introduced by Egenhofer. We show that the closure of these relationships under appropriate logical operators yields languages which are complete for topological prope...
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 ..."
Abstract

Cited by 36 (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.
Topological Relations between Regions in R² and Z²
, 1993
"... Users of geographic databases that integrate spatial data represented in vector and raster models, should not perceive the differences among the data models in which data are represented, nor should they be forced to apply different concepts depending on the model in which spatial data are repre ..."
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Cited by 29 (2 self)
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Users of geographic databases that integrate spatial data represented in vector and raster models, should not perceive the differences among the data models in which data are represented, nor should they be forced to apply different concepts depending on the model in which spatial data are represented. A crucial aspect of spatial query languages for such integrated systems is the need mechanisms to process queries about spatial relations in a consistent fashion. This paper compares topological relations between spatial objects represented in a continuous (vector) space of ## and a discrete (raster) space of ZZ . It applies the 9intersection, a frequently used formalism for topological spatial relations between objects represented in a vector data model, to describe topological relations for bounded objects represented in a raster data model. We found that the set of all possible topological relations between regions in ## is a subset of the topological relations that can be realized between two bounded, extended objects in ZZ . At a
Querying Spatial Databases via Topological Invariants
 In PODS'98
, 1998
"... The paper investigates the use of topological annotations (called topological invariants) to answer topological queries in spatial databases. The focus is on the translation of topological queries against the spatial database into queries against the topological invariant. The languages considered ..."
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Cited by 17 (2 self)
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The paper investigates the use of topological annotations (called topological invariants) to answer topological queries in spatial databases. The focus is on the translation of topological queries against the spatial database into queries against the topological invariant. The languages considered are firstorder on the spatial database side, and fixpoint + counting, fixpoint, and firstorder on the topological invariant side. In particular, it is shown that fixpoint + counting expresses precisely all the ptime queries on topological invariants; if the regions are connected, fixpoint expresses all ptime queries on topological invariants. 1 Introduction Spatial data is an increasingly important part of database systems. It is present in a wide range of applications: geographic information systems, video databases, medical imaging, CADCAM, VLSI, robotics, etc. Different applications pose different requirements on query languages and therefore on the kind of spatial information th...
The Complexity of Reasoning about Spatial Congruence
 Journal of Artificial Intelligence Research
, 1999
"... In the recent literature of Artificial Intelligence, an intensive research effort has been spent, for various algebras of qualitative relations used in the representation of temporal and spatial knowledge, on the problem of classifying the computational complexity of reasoning problems for subset ..."
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Cited by 12 (3 self)
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In the recent literature of Artificial Intelligence, an intensive research effort has been spent, for various algebras of qualitative relations used in the representation of temporal and spatial knowledge, on the problem of classifying the computational complexity of reasoning problems for subsets of algebras. The main purpose of these researches is to describe a restricted set of maximal tractable subalgebras, ideally in an exhaustive fashion with respect to the hosting algebras. In this paper we introduce a novel algebra for reasoning about Spatial Congruence, show that the satisfiability problem in the spatial algebra MC4 is NPcomplete, and present a complete classification of tractability in the algebra, based on the individuation of three maximal tractable subclasses, one containing the basic relations. The three algebras are formed by 14, 10 and 9 relations out of 16 which form the full algebra. 1. Introduction Qualitative spatial reasoning has received an increasin...
Towards Usable Topological Operators at GIS User Interfaces
 In Toppen, F. and P. Prastacos (eds). 7 th Conference on Geographic Information Science
, 2004
"... The topological relations defined by formal mathematical models like the nineintersection are not adequate for user interaction. Human subject tests have shown that people rather use less, grouped, and overlapping relations. But these are only first results and a theory for building cognitively ade ..."
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Cited by 1 (1 self)
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The topological relations defined by formal mathematical models like the nineintersection are not adequate for user interaction. Human subject tests have shown that people rather use less, grouped, and overlapping relations. But these are only first results and a theory for building cognitively adequate operator sets is missing. This is reflected by the considerably different operator sets of current software products. So far, human subject tests concerning cognitive adequacy of topological operators have followed a general approach detached from human activities. We propose to change the focus and examine which topological relations humans use when accomplishing tasks. The