Model For Topological Relations Research Articles

Formal Verification of a Topological Spatial Relations Model for Geographic Information Systems in Coq

Geographic information systems have undergone rapid growth for decades. Topology has provided valuable modeling tools in the development of this field. Formal verification ofthe model of topological spatial relations can provide a reliable guarantee for the correctness of geographic information systems. We present a proof of the topological spatial relations model that has been formally verified in the Coq proof assistant. After an introduction to the formalization of the axiomatic set theory of Morse–Kelley, the formal description of the elementary concepts and properties of general topology is developed. The topological spatial relations between two sets are described by using the concept of the intersection value. Finally, we formally proved the topological spatial relations between two sets which are restricted to the regularly closed and the planar spatial regions. All the proof details are strictly completed in Coq, which shows that the correctness of the theoretical model for geographic information systems can be checked by a computer. This paper provides a novel method to verify the correctness of the topological spatial relations model. This work can also contribute to the creation and validation of various geological models and software.

A Generalized 9-Intersection Model for Topological Relations between Regions with Holes

Current models cannot distinguish detailed topological relations between regions with holes. In order to solve this problem, a new detailed representation model for topological relations is proposed in this study. In this model, the key is to describe topological relations caused by multi-holes. Definitions of regions with holes and real objects expressed by elements of regions with holes are comprehensively analyzed, and then a practical definition of regions with holes is presented. Based on the 9-intersection model, a generalized 9-intersection model is proposed, which can completely describe detailed topological relations between regions with holes. For showing the description ability, the model is applied to describe detailed topological relations between regions with holes and detailed topological relations between objects of different complexities with the same major categories of topological relations, and a case study is designed to show the practicality of the model. The results show that the model is valid.

Topological relations between directed line segments in the cyclic space

Topological relations between directed line segments (DLs) may contribute to queries and analyses related to noninstantaneous phenomena whose position changes over time. Although considerable research has been conducted to study topological relation models and the specification of the topological relations that exist in reality, further work is required to consider what types of topological relations between DLs in a cyclic space can be realized. This research is a contribution to the clarification of the topological relations between DLs in a cyclic space that can occur in reality. A DL is divided into four primitives: a starting point, an ending point, an interior, and an exterior. A topological relation model between two DLs in a cyclic space with a 4 × 4 matrix is proposed in this article. A total of 38 topological relations between DLs in the cyclic space are distinguished, and the matrix patterns and the corresponding geometric interpretations of the 38 topological relations are shown to prove the existence of the topological relations. Eleven negative conditions are summarized to prove the completeness of the 38 topological relations. The cyclic interval relations, spherical topological relations, and topological relations presented in this research are compared. The results show the following: (1) the proposed topological relation model can well represent the topological relations between DLs, and (2) the proposed 11 negative conditions can be used to prove the completeness of the 38 topological relations.

Discovery of topological constraints on spatial object classes using a refined topological model

In a typical data collection process, a surveyed spatial object is annotated upon creation, and is classified based on its attributes. This annotation can also be guided by textual definitions of objects. However, interpretations of such definitions may differ among people, and thus result in subjective and inconsistent classification of objects. This problem becomes even more pronounced if the cultural and linguistic differences are considered. As a solution, this paper investigates the role of topology as the defining characteristic of a class of spatial objects. We propose a data mining approach based on frequent itemset mining to learn patterns in topological relations between objects of a given class and other spatial objects. In order to capture topological relations between more than two (linear) objects, this paper further proposes an extension of the 9-intersection model for topological relations of line geometries. The discovered topological relations form topological constraints of an object class that can be used for spatial object classification. A case study has been carried out on bridges in the OSM dataset for the state of Victoria, Australia. The results show that the proposed approach can successfully learn topological constraints for the class bridge, and that the proposed extended topological model for line geometries outperforms the 9-intersection model in this task.

Detailed model of topological and metric relationships between a line and region

Topological relation is one of the basic topics of geographic information systems (GISs), and it has been widely applied in data organization and spatial analysis. It seems whether a simple topological relation can hardly represent complex space situations in some cases. Different topological relations may result in the same expression by some topological relation models, among which the 9-Intersection Model (9IM), and its extended models are well known. In this paper, a detailed model and calculation method for topological relations are proposed, to describe the complex topological relations between lines and regions more exactly. It is better to answer the question, e.g., “How does the road cross the park?” The programs are developed to compute the complex topological relations between a line and region, and the actual area of land use parcels, from which the area of related linear features wait to be subtracted.

A distance-based topological relation model between spatial regions

Although the definitions of formal models used to represent spatial relations have gained increasing attention over the past 30 years, the linkage between topology and distance has not yet been effectively established. A topological relation model called the distance-based topological relation model (D-TRM) that considers both the topology and distance of spatial regions is proposed. The D-TRM is divided into two subtypes: the actual DTRM (AD-TRM) and the signed DTRM (SD-TRM). The actual distance is based on the distance in a two-dimensional space. The signed distance is based on the sign of the actual distance. Eight topological relations, namely, disjoint, meet, overlap, cover, contain, equal, coveredBy and inside, represented by the AD-TRM and SD-TRM are shown. The mutual exclusiveness among these eight topological relations represented by the SD-TRM is proven. The topological relation representations from the 9-intersection model (9IM), the splitting measures of the 9IM (SP-9IM), the SD-TRM and the AD-TRM are discussed, and the interoperability of each of the above models is summarised. The topological relation representation between the AD-TRM and the comprehensive set of 11 metric refinements is compared. The results show the following: (1) as the generalisation of the AD-TRM, the SD-TRM can concisely represent the topological relations; (2) the topology and distance between two spatial regions can be represented by the AD-TRM in a unified framework; (3) the AD-TRM provides a greater level of detail than the 9IM and (4) the D-TRM can express more distance information than the comprehensive set of 11 metric refinements.

Classification of topological relations between spatial objects in two‐dimensional space within the dimensionally extended 9‐intersection model

AbstractAs an important topological relation model, the dimensionally extended 9‐intersection model (DE‐9IM) has been widely used as a basis for standards of queries in spatial databases. However, the negative conditions for the specification of the topological relations within the DE‐9IM have not been studied. The specification of the topological relations is closely related to the definition of the spatial objects and the topological relation models. The interior, boundary, and exterior of the spatial objects, including the point, line, and region, are defined. Within the framework of the DE‐9IM, 43 negative conditions are proposed to eliminate impossible topological relations. Configurations of region/region, region/line, line/line, region/point, line/point, and point/point relations are drawn. The mutual exclusion of the negative conditions is discussed, and the topological relations within the framework of 9IM and DE‐9IM are compared. The results show that: (1) impossible topological relations between spatial objects can be eliminated by the application of 43 negative conditions; and (2) 12 relations between two regions, 31 relations between a region and a line, 47 relations between two lines, three relations between a region and a point, three relations between a line and a point, and two relations between two points can be distinguished by the DE‐9IM.

A Model of Fuzzy Topological Relations for Simple Spatial Objects in GIS

The goal of this paper is to present a new model of fuzzy topological relations for simple spatial objects in Geographic Information Sciences (GIS). The concept of computational fuzzy topological space is applied to simple fuzzy objects to efficiently and more accurately solve fuzzy topological relations, extending and improving upon previous research in this area. Firstly, we propose a new definition for simple fuzzy line segments and simple fuzzy regions based on computational fuzzy topology. And then, we also propose a new model to compute fuzzy topological relations between simple spatial objects, an analysis of the new model exposes:(1) the topological relations of two simple crisp objects; (2) the topological relations between one simple crisp object and one simple fuzzy object; (3) the topological relations between two simple fuzzy objects. In the end, we have discussed some examples to demonstrate the validity of the new model, through an experiment and comparisons of existing models, we showed that the proposed method can make finer distinctions, as it is more expressive than the existing fuzzy models.

Model of Automatic Distribution of Topologically Related Objects to Layers in Navigation Systems

In article the problem of automatic distribution of spatial objects to layers is considered. The result of the distribution is multi-layered vector map. Hierarchical model of topological relations has been developed. It is the basis for implementation of objects distribution algorithms. The practical significance of the work is processing of spatial objects in navigation systems.

Developing Data Fusion Method for Indoor Space Modeling based on IndoorGML Core Module

응용프로그램은 그 목적에 따라 최적의 데이터 모델을 활용하며, 이러한 응용프로그램을 위한 3차원 모델링 데이터는 선택된 데이터 모델을 기반으로 생성된다. 이러한 이유로, 동일한 공간의 지형지물을 표현하기 위해 다양한 데이터 셋이 존재한다. 그러한 중복된 데이터 셋은 공간정보 산업의 재정적 측면에서 문제를 가져올 뿐만 아니라, 시스템호환성과 데이터 비교가능성에서도 심각한 문제를 야기한다. 이러한 문제를 극복하기 위하여, 본 연구에서는 항목클래스내의 공간객체들 간의 위상적 관계를 이용하여 TRM (Topological Relation Method)이라고 하는 공간데이터융합 방법을 제안한다. TRM은 응용프로그램 수준에서 구현되는 공간데이터 융합방법으로써, 서로 다른 데이터 모델에 의해 생성된 기하데이터들을 응용시스템에서 어떠한 데이터 변환이나 교환 과정을 거치지 않고, 직접적으로 실내공간 위치기반 서비스에 제공하기 위해 사용된다. 이러한 위상관계는 IndoorGML의 기본 개념으로 정의 및 기술된다. TRM의 개념을 기술한 후, 3D GIS상에서 제안된 데이터 융합방법의 실험적 구현을 보여준다. 마지막으로서 본 연구의 한계와 향후 연구에 대해 정리한다. According to the purpose of applications, the application program will utilize the most suitable data model and 3D modeling data would be generated based on the selected data model. In these reasons, there are various data sets to represent the same geographical features. The duplicated data sets bring serious problems in system interoperability and data compatibility issues, as well in finance issues of geo-spatial information industries. In order to overcome the problems, this study proposes a spatial data fusion method using topological relationships among spatial objects in the feature classes, called Topological Relation Model (TRM). The TRM is a spatial data fusion method implemented in application-level, which means that the geometric data generated by two different data models are used directly without any data exchange or conversion processes in an application system to provide indoor LBSs. The topological relationships are defined and described by the basic concepts of IndoorGML. After describing the concepts of TRM, experimental implementations of the proposed data fusion method in 3D GIS are presented. In the final section, the limitations of this study and further research are summarized.

Representation, reasoning and similar matching for detailed topological relations with DTString

Most current topological relation models cannot capture the details of spatial relations; thus, complex spatial relations cannot be distinguished using these models. Under certain circumstances, only detailed topological relations can satisfy users’ demand. To resolve this problem, this study proposes a new topological relation model named DTString that describes the full details of the topological relation between two regions by a boundary string. DTString is proved to be a JEPD (Jointly Exhaustive and Pairwise Disjoint) and atomic relation model. Thus, undividable topological relations can be captured with it, and this characterization makes it suitable for modeling complex topological relations. Furthermore, DTString-based reasoning algorithms are investigated; they are more efficient than algorithms based on geometric calculations because they use purely string-based calculations. Finally, methods for retrieving similar geometrical structures are investigated by combining topological relations with shape and distribution features. Experiment results show that DTString outperforms existing topological models. DTString and the proposed similarity measurements can be potentially applied to content-based image retrieval, spatial query and CAD, etc.

The Reasoning and Application Model for Topological Relations between a Concave Region and a Simple Broad Boundary Region

According to the nature of 16 - the intersection model and the constraints of the broad boundary region, get 67 topological relations that can be realized between a concave region and a simple broad boundary region. We realize the reasoning of topological relations between a concave region and a simple broad boundary region, and establish topology model of typhoon rainfall warning, and provide reference for expand of the related topology model applications.

The General Description of Topological Relations based on Node Degree and Euler-number

Topological relationship is one of the most basic and the most important relations between two spatial objects in GIS.However,current topological models have deficiencies when they are used to describe detailed topological relations.In this paper,one general descriptive form of topological relation model was proposed based on the node degree and the euler-number aiming at random two objects.In this model,all parts of one spatial object were treated as a whole object,and intersection difference operators were selected from several set operators to distinguish the topological relations,and four types of topological invariants were used for the computational results of contents,dimension,node degree,and euler-number.The topological description form of line-line,line-area,area-area were discussed emphatically.

Topological Relations between Convex Regions

Topological relations between spatial objects are the most important kind of qualitative spatial information. Dozens of relation models have been proposed in the past two decades. These models usually make a small number of distinctions and therefore can only cope with spatial information at a fixed granularity of spatial knowledge. In this paper, we propose a topological relation model in which the topological relation between two convex plane regions can be uniquely represented as a circular string over the alphabet {u; v; x; y}. A linear algorithm is given to compute the topological relation between two convex polygons. The infinite relation calculus could be used in hierarchical spatial reasoning as well as in qualitative shape description.

The Galois lattice as a hierarchical structure for topological relations

This paper presents the construction and the comparison of Galois lattices of topological relations for qualitative spatial representation and reasoning. The lattices rely on a correspondence between computational operations working on quantitative data, on the one hand, and topological relations working on qualitative knowledge units, on the other hand. After introducing the context of the present research work, i.e. the RCC-8 model of topological relations, we present computational operations for checking topological relations on spatial regions. From these operations are derived two sets of computational conditions that can be associated to topological relations through a Galois connection. The associated Galois lattices are presented and compared. Elements on the practical use of the lattices for representing spatial knowledge and for reasoning are also introduced and discussed.

The four intersection-and-difference model for line-line topological relations

The description of line-line topological relations is still an unsolved issue although much effort has been done. The problem is involved in many practical applications such as spatial query, spatial analysis and cartographic generalization. To develop a sound and effective approach to describe line-line relations, it is first necessary to define the topology of an individual line, i.e., local topology. The concept of connective degree is used for the identification of topological differences in the geometric structure of a line. The general topological definition of a line is given, i.e., endpoints set and interior point set. This definition can be applied to the embedded spaces of different dimensions, whether co-dimension is equal to or larger than zero. On this basis, a generic model called the 4 intersection-and-difference is set up for the description of basic line-line topological relations, upon which a conceptual neighborhood graph is built with consideration of topological distance. It is concluded that the proposed model can represent the property of topological changes, and basic relations between line segments in IR1 and IR2.

Extended model of topological relations between spatial objects in geographic information systems

This paper presents an extended model for describing topological relations between two sets (objects) in geographic information systems (GIS). First, based on the definition of the topological relations between two objects, we uncover a sequence of topological relations between two convex sets.Second, an extended model for topological relations between two sets is proposed based on the new definition. The topological relations between two convex sets are expressed as a sequence of 4×4 matrices, which are the topological properties of Ao∩Bo, Ao\\B, Bo\\A, ∂A∩∂B. The model is also extended for handling the properties of the topological relations between two non-convex sets, where the factor of first fundamental group is added to A∪B to handle these complex relations.The results show that the number of topological relations between the two sets is not as simple as finite but infinite and can be approximated by a sequence of matrices.

A spatial model for complex objects with a broad boundary supporting queries on uncertain data

This paper proposes a new data model for spatial databases that supports spatial entities with a broad boundary. The model is proposed as a solution to the problem of handling uncertainty in spatial data. Broad boundaries absorb all the uncertainty by which spatial data are commonly affected and allow computations in presence of uncertainty without rough simplifications of the reality. A specific model for topological relations between regions with a broad boundary is then developed and hierarchically structured topological operators are proposed. Such operators allow users to progressively refine their queries and constitute the basis for a spatial extension to SQL. The model proposed in the paper can be easily integrated into existing data models for spatial databases.

Metric details for natural-language spatial relations

Spatial relations often are desired answers that a geographic information system (GIS) should generate in response to a user's query. Current GIS's provide only rudimentary support for processing and interpreting natural-language-like spatial relations, because their models and representations are primarily quantitative, while natural-language spatial relations are usually dominated by qualitative properties. Studies of the use of spatial relations in natural language showed that topology accounts for a significant portion of the geometric properties. This article develops a formal model that captures metric details for the description of natural-language spatial relations. The metric details are expressed as refinements of the categories identified by the 9-intersection, a model for topological spatial relations, and provide a more precise measure than does topology alone as to whether a geometric configuration matches with a spatial term or not. Similarly, these measures help in identifying the spatial term that describes a particular configuration. Two groups of metric details are derived: splitting ratios as the normalized values of lengths and areas of intersections; and closeness measures as the normalized distances between disjoint object parts. The resulting model of topological and metric properties was calibrated for 64 spatial terms in English, providing values for the best fit as well as value ranges for the significant parameters of each term. Three examples demonstrate how the framework and its calibrated values are used to determine the best spatial term for a relationship between two geometric objects.

On the equivalence of topological relations

Abstract The 4-intersection, a model for binary topological relations, is based on the intersections of the boundaries and interiors of two point sets in a topological space, considering the content invariant (i.e., emptiness/non-emptiness) of the intersections. If the 4-intersections of two pairs of point sets have different contents, then their topological relations are different as well; however, the reverse cannot be stated as there may be different topological relations that map onto a 4-intersection with the same content. This paper refines the model of empty/non-empty 4-intersections with further topological invariants to account for more details about topological relations. The invariants used are the dimension of the components, their types (touching, crossing, and different refinements of crossings), their relationships with respect to the exterior neighbourhoods, and the sequence of the components. These invariants, applied to non-empty boundary-boundary intersections, comprise a classification...