Eliseo Clementini

A Small Set of Formal Topological Relationships Suitable for End-User Interaction

Topological relationships between spatial objects represent important knowledge that users of geographic information systems expect to retrieve from a spatial database. A difficult task is to assign precise semantics to user queries involving concepts such as “crosses”, “is inside”, “is adjacent”. In this paper, we present two methods for describing topological relationships. The first method is an extension of the geometric point-set approach by taking the dimension of the intersections into account. This results in a very large number of different topological relationships for point, line, and area features. In the second method, which aims to be more suitable for humans, we propose to group all possible cases into a few meaningful topological relationships and we discuss their exclusiveness and completeness with respect to the point-set approach.

Qualitative representation of positional information

Abstract A framework for the qualitative representation of positional information in a two-dimensional space is presented. Qualitative representations use discrete quantity spaces, where a particular distinction is introduced only if it is relevant to the context being modeled. This allows us to build a flexible framework that accommodates various levels of granularity and scales of reasoning. Knowledge about position in large-scale space is commonly represented by a combination of orientation and distance relations, which we express in a particular frame of reference between a primary object and a reference object. While the representation of orientation comes out to be more straightforward, the model for distances requires that qualitative distance symbols be mapped to geometric intervals in order to be compared; this is done by defining structure relations that are able to handle, among others, order of magnitude relations; the frame of reference with its three components (distance system, scale, and type) captures the inherent context dependency of qualitative distances. The principal aim of the qualitative representation is to perform spatial reasoning: as a basic inference technique, algorithms for the composition of positional relations are developed with respect to same and different frames of reference. The model presented in this paper has potential applications in areas as diverse as Geographical Information Systems (GIS), Computer Aided Design (CAD), and Document Recognition.

Topological relations between regions with holes

The 4-intersection, a model for the representation of topological relations between 2-dimensional objects with connected boundaries and connected interiors, is extended to cover topological relations between 2-dimensional objects with arbitrary holes, called regions with holes. Each region with holes is represented by its generalized region—the 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 1-spheres. 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 multi-level 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.

Approximate topological relations

Abstract In spatial data models for various applications, such as geographical information systems (GISs), the importance of topological relations is widely recognized. Topology makes very general statements about the structure and the relations of spatial objects. A refinement of topology by means of other geometric aspects can help to bend the various models that have been developed for topological relations towards a more effective description of geographic space. The introduction of broad boundaries is a direction to define approximate topological relations between spatial objects. In this paper, approximate topological relations are destined to capture boundary uncertainty, variations over time, proximity measures, and vector-raster representations. Approximate topological relations are structured in conceptual neighborhood graphs that have a twofold interpretation: two neighboring relations are at topological distance 1 in terms of the nine-intersection model and can be obtained, one from the other, by an elementary continuous deformation.

A model for representing topological relationships between complex geometric features in spatial databases

Various models for the representation of topological relationships have been developed. The aim of this paper is to show that the set of relationships proposed in [7] (the CBM), for describing topological relationships among two-dimensional simple features, is applicable with few modifications to the case of complex features (that is, areas made up of several components possibly containing holes, lines with self-intersections, and/or more than two endpoints, and so on). The CBM offers a small set of topological relationships with high expressiveness which is proven to be mutually exclusive and complete, and therefore suitable to be embedded in a spatial query language.

A comparison of methods for representing topological relationships

In the field of spatial information systems, a primary need is to develop a sound theory of topological relationships between spatial objects. A category of formal methods for representing topological relationships is based on point-set theory. In this paper, a high level calculus-based method is compared with such point-set methods. It is shown that the calculus-based method is able to distinguish among finer topological configurations than most of the point-set methods. The advantages of the calculus-based method are the direct use in a calculus-based spatial query language and the capability of representing topological relationships among a significant set of spatial objects by means of only five relationship names and two boundary operators.

Composite regions in topological queries

Spatial data are at the core of many scientific information systems. The design of suitable query languages for spatial data retrieval and analysis is still an issue on the cutting edge of research. The primary requirement of these languages is to support spatial operators. Unfortunately, current systems support only simplified abstractions of geographic objects based on simple regions which are usually not sufficient to deal with the complexity of the geographic reality. Composite regions, which are regions made up of several components, are necessary to overcome those limits. The paper introduces a two-level formal model suitable for representing topological relationships among composite regions. The contribution gives the needed formal background for adding composite regions inside a spatial query language with the purpose of answering topological queries on complex geographic objects.

Integration of Imperfect Spatial Information

The theme of this paper is integration of information arising from observations of spatial entities and relationships. The assumption is that observations are imperfect; in particular, that they are imprecise and inaccurate. Each observation is made in a context that among other things provides a level of resolution. So, a treatment of integration of observations of this type must take account of multiresolution spatial data models. After an introduction, the paper discusses an ontology of imperfection, focusing on imprecision and inaccuracy. The paper goes on to consider logics that are appropriate for integration of information arising from imperfect observations. Two case studies, showing some of the facets of this treatment are developed in greater detail. The first case study considers integration of imperfect (inaccurate and imprecise) observations of a single spatial region. The second case study develops the theory of regions with broad boundary to address the issue of integrating imprecise observations of spatial relationships. ( 2001 Academic Press

Mining multiple-level spatial association rules for objects with a broad boundary

Abstract Spatial data mining, i.e., mining knowledge from large amounts of spatial data, is a demanding field since huge amounts of spatial data have been collected in various applications, ranging from remote sensing to geographical information systems (GIS), computer cartography, environmental assessment and planning. The collected data far exceeds peoples ability to analyze it. Thus, new and efficient methods are needed to discover knowledge from large spatial databases. Most of the spatial data mining methods do not take into account the uncertainty of spatial information. In our work we use objects with broad boundaries, the concept that absorbs all the uncertainty by which spatial data is commonly affected and allows computations in the presence of uncertainty without rough simplifications of the reality. The topological relations between objects with a broad boundary can be organized into a three-level concept hierarchy. We developed and implemented a method for an efficient determination of such topological relations. Based on the hierarchy of topological relations we present a method for mining spatial association rules for objects with uncertainty. The progressive refinement approach is used for the optimization of the mining process.

Topological invariants for lines

A set of topological invariants for relations between lines embedded in the 2-dimensional Euclidean space is given. The set of invariants is proven to be necessary and sufficient to characterize topological equivalence classes of binary relations between simple lines. The topology of arbitrarily complex geometric scenes is described with a variation of the same set of invariants. Polynomial time algorithms are given to assess topological equivalence of two scenes. Invariants and efficient algorithms is due to application areas of spatial database systems where a model for describing topological relations between planar features is sought.