Spatial database systems and Geographic Information Systems (GIS) commonly assume that spatial objects have crisp boundaries and homogenous interiors and thus offer crisp spatial type systems to applications. However, many real-world spatial phenomena inherently exhibit a vague nature and structure that prevents the precise definition of their boundaries and/or interiors. A spatial object is then called fuzzy since certain locations cannot be unambiguously and completely assigned either to the object or to its complement. This paper gives an overview and some highlights of an abstract, formal, and conceptual type system called Fuzzy Spatial Algebra (FUSA). Based on fuzzy set theory and fuzzy topology, FUSA provides a collection of fuzzy spatial data types for fuzzy points, fuzzy lines, and fuzzy regions in the two-dimensional Euclidean space. FUSA also offers a set of expressive spatial operations such as fuzzy union, fuzzy intersection, and fuzzy difference to perform geometric computations on fuzzy spatial objects and ensures the closure property of creating valid and consistent objects of a fuzzy spatial data type. FUSA serves as the formal specification for implementing fuzzy spatial type systems in spatial databases and GIS, hence enhancing the capabilities of many applications.

Increasingly, geoscientists and spatial data scientists have shown interest in modeling and analyzing spatial phenomena characterized by the feature of spatial fuzziness. Applying fuzzy logic and fuzzy inference methods to fuzzy spatial objects leads to fuzzy spatial inference models. These models pursue the goal of discovering new meaningful findings from fuzzy spatial data, hence contributing to data knowledge discovery and sharing this goal with spatial data science. In this paper, we introduce a novel type of inference method called region inference; it combines spatial query processing with fuzzy inference methods. The objective is to capture all points that intersect a search object (e.g., a query window) and whose output values fulfill some specific user requirements (e.g., the points with the maximum or minimum inferred values). For this, we propose, evaluate, and compare query window inference methods in fuzzy spatial inference models. In addition, we show their characterization and applicability in real spatial applications.

Spatial data science emerges as an important subclass of data science and focuses on extracting meaningful information and knowledge from spatial data to enable effective communication and interpretation of both spatial data and analytic results. It emphasizes the importance of location and spatial interaction by storing, analyzing, retrieving, and visualizing spatial and geometric information. Frequently, spatial objects are afflicted by spatial fuzziness, characterizing spatial objects with blurred interiors, uncertain boundaries, and imprecise locations. Fuzzy set theory and fuzzy logic have become powerful tools to adequately represent spatial fuzziness. This paper provides a survey and a review of the literature to understand the application of fuzzy approaches to spatial data science (projects) with the objective of proposing, motivating, and envisioning fuzzy spatial data science.

GIS and spatial data science (SDS) tools have been recently approaching each other by establishing bridge technologies between them. R as one of the most prominent programming languages used in SDS projects has been granted access to GIS infrastructure, while R scripts can be integrated and executed in GIS functions. Unfortunately, the treatment of spatial fuzziness has so far not been considered in SDS projects and bridge technologies due to a lack of software packages that can handle fuzzy spatial objects. This paper introduces an R package named fsr as an implementation of the fuzzy spatial data types, operations, and predicates of the Spatial Plateau Algebra that is based on the abstract Fuzzy Spatial Algebra. This R package solves the problem of constructing fuzzy spatial objects as spatial plateau objects from real datasets and describes how to conduct exploratory spatial data analysis by issuing geometric operations and topological predicates on fuzzy spatial objects. Further, fsr provides the possibility of designing fuzzy spatial inference models to discover new findings from fuzzy spatial objects. It optimizes the inference process by deploying the particle swarm optimization to obtain the point locations with the maximum or minimum inferred values that answer a specific user request.

Fuzzy set theory has found increasing interest in the geosciences, geographic information systems, and spatial database systems to represent geometric objects in the two-dimensional space that reveal an intrinsically vague or fuzzy nature. A spatial object is fuzzy if it contains locations that cannot be assigned completely to the object or to its complement. From a conceptual perspective, fuzzy spatial data types for 0-dimensional fuzzy points, 1-dimensional fuzzy lines, and 2-dimensional fuzzy regions in the plane have been introduced, e.g., by the authors formal Fuzzy Spatial Algebra (FUSA). But the limitation of fuzzy spatial objects to a fixed geometric dimension turns out to be sometimes too restrictive since such objects could benefit from a characterization in terms of several fuzzy spatial sub-objects of different dimensions. An example is a river that consists of 1-dimensional linear parts and 2-dimensional areal parts. For this purpose, this paper introduces a new fuzzy spatial composition type with corresponding operations. It allows one to accommodate fuzzy spatial sub-objects that are either adjacent or disjoint. As a generalization of this type, this paper provides a fuzzy spatial collection type with corresponding operations. Fuzzy collection objects allow one to keep an arbitrary, finite number of fuzzy spatial objects of possibly different dimensions without any topological constraints in a single object. Application examples show how these new data types can be deployed.

Fuzzy set theory has found increasing interest in the geosciences, geographic information systems, and spatial database systems to represent objects in the two-dimensional space that are afflicted with spatial fuzziness. From a system perspective, fuzzy spatial data types for fuzzy points, fuzzy lines, and fuzzy regions have been introduced, e.g., by the authors' formal Fuzzy Spatial Algebra (FUSA). The authors' Spatial Plateau Algebra (SPA) provides an implementation of FUSA by means of spatial plateau data types for plateau point, plateau line, and plateau region objects. It is based on well known non-fuzzy, crisp spatial data types. In this paper, we deal with the issue of constructing fuzzy region objects as plateau region objects from real point sets by leveraging domain expert knowledge and by assuming that each point is assigned a numerical value of a given application context. For this, we propose a general two-stage data extraction method. The first stage deploys a fuzzification policy to assign membership degrees to the points in the set. Examples of such policies are the execution of fuzzy clustering algorithms or the usage of fuzzy sets. The second stage uses a construction policy to build plateau regions by using the membership degrees generated in the first stage as input. Examples of such construction policies are the computation of fuzzy Voronoi diagrams or fuzzy Delaunay triangulations, or the calculation of fuzzy convex hulls.

Many geographical applications have to deal with spatial objects that reveal an intrinsically vague or fuzzy nature. A spatial object is fuzzy if locations exist that cannot be assigned completely to the object or to its complement. Spatial database systems and Geographical Information Systems are currently unable to cope with this kind of data. Based on an available abstract data model of fuzzy spatial data types for fuzzy points, fuzzy lines, and fuzzy regions, this paper proposes the Spatial Plateau Algebra that provides spatial plateau data types as an executable type system for fuzzy spatial data types. Each spatial plateau object consists of a finite number of crisp counterparts that are all adjacent or disjoint to each other, are associated with different membership values, and hence form different plateaus. The formal framework and the implementation are based on well known, exact models and available implementations of crisp spatial data types. Spatial plateau set and metric operations as well as spatial plateau topological predicates on spatial plateau objects are expressed as a combination of geometric methods on the underlying crisp spatial objects. This paper offers a conceptually clean specification for implementing a database extension for fuzzy spatial objects and their operations and predicates. Further, we demonstrate the embedding of these new data types as attribute data types in a database schema as well as the incorporation of fuzzy spatial operations and predicates into a database query language.

Spatial Decision Support Systems have received increasing interest in geographical, political, and economical applications such as agricultural cultivation, disaster management, and industrial settlement. For instance, farmers want to know what the best farmland areas are to grow a specific crop, political decision makers want to know what the areas are that should be protected based on risk zones, and companies would like to know the best location to place a new production facility. In many cases, the spatial phenomena of interest have a vague and imprecise extent and can be adequately represented by fuzzy spatial objects such as fuzzy regions. In this paper, we formally propose a general-purpose model named Fuzzy Inference on Fuzzy Spatial Objects (FIFUS) that incorporates fuzzy spatial objects into its inference strategy and supplies the user with recommendations, estimations, and predictions based on fuzzy inference rules and expert knowledge.

Nowadays Geographical Information Systems (GIS) and spatial database systems are mainly able to handle crisp spatial objects, i.e., objects in space whose locations, extents, shapes, and boundaries are precisely determined. However, geoscientists and advanced GIS applications are increasingly interested in handling spatial objects that are characterized by the feature of spatial fuzziness and do not have sharp but blurred boundaries and/or interiors; hence, we call them fuzzy spatial objects. Examples are air polluted areas and temperature zones. In the same way as fuzzy spatial objects are blurred, the topological relationships (e.g., overlap, inside) between them are blurred too. In this conceptual paper, we propose a novel model to formally define fuzzy topological relationships for simple and complex fuzzy regions. Such a fuzzy relationship computes a membership degree between 0 and 1 that indicates to which extent this relationship holds. It is mapped to a high-level fuzzy modifier and transformed into a Boolean predicate that can be embedded into spatial queries.

Decision support based on spatial (and not only alphanumerical) data has received increasing interest in geographical applications, such as geoscience, agriculture, and economics applications, and has led to Spatial Decision Support Systems (SDSS). SDSS use spatial database systems and Geographical Information Systems as their data management and analysis components in order to get and handle the needed spatial data and perform recommendations, estimations, or predictions. For instance, farmers want to know what the best areas of their farmland are to grow a specific crop. In most cases, the extent and the properties of the spatial phenomena of interest are vague and imprecise. They can be adequately represented by fuzzy spatial objects (e.g., fuzzy points, fuzzy lines, fuzzy regions). In this paper, we formally propose a model named Fuzzy Inference on Fuzzy Spatial Objects (FIFUS), which infers recommendations, estimations, and predictions based on fuzzy rules and knowledge of domain specialists. It incorporates fuzzy spatial objects into the components of the existing fuzzy inference methods in order to take into account the spatial imprecision found in the real world. As a main advantage, FIFUS is a general-purpose model and can thus be applied in many geoscience applications.

In recent decades, applications like geospatial, genomic, and multimedia have made use of very large and diverse application objects such as spatial networks and protein structures. These objects are complex in the sense that they are highly structured and of variable size. Storing, accessing and manipulating them in a standard and efficient manner is very challenging. The state-of-the-art solutions handle such objects by using file system formats like HDF and XML, serialization technique like Protocol Buffers and BLOB data type in databases. However, specialized file format solutions lack any well established database system features, and neither a uniform concept nor mechanisms exist for supporting complex objects for BLOBs. In this article, a novel and database-friendly framework of specifying and interpreting complex objects is proposed. Empirical studies have shown that our approach outperforms prevailing methods with efficient processing time and less storage consumption.

Spatial database systems and Geographical Information Systems (GIS) are currently only able to handle crisp spatial objects, i.e., objects whose extent, shape, and boundary are precisely determined. However, GIS applications are also interested in managing vague or fuzzy spatial objects. Spatial fuzziness captures the inherent property of many spatial objects in reality that do not have sharp boundaries and interiors or whose boundaries and interiors cannot be precisely determined. While topological relationships have been broadly explored for crisp spatial objects, this is not the case for fuzzy spatial objects. In this paper, we propose a novel model to formally define fuzzy topological predicates for simple and complex fuzzy regions. The model encompasses six fuzzy predicates (overlap, disjoint, inside, contains, equal and meet), wherein here we focus on the fuzzy overlap and the fuzzy disjoint predicates only. For their computation we consider two low-level measures, the degree of membership and the degree of coverage, and map them to high-level fuzzy modifiers and linguistic values respectively that are deployed in spatial queries by end-users.

Data warehouses help to store and analyze large multidimensional datasets and provide enterprise decision support. With an increased availability of spatial data in recent years, several new strategies have been proposed to enable their integration i

Besides topological relationships and approximate relationships, cardinal directions like north and southwest have turned out to be an important class of qualitative spatial relationships. They are of interdisciplinary interest in fields like cognitive science, robotics, artificial intelligence, and qualitative spatial reasoning. In spatial databases and Geographic Information Systems (GIS) they are frequently used as join and selection criteria in spatial queries. However, the available computational models of cardinal directions suffer a number of problems like the use of too coarse approximations of the two spatial operand objects in terms of single representative points or minimum bounding rectangles, the lacking property of converseness of the cardinal directions computed, and the limited applicability to simple instead of complex regions only. This article proposes and formally defines a novel two-phase model, called the Objects Interaction Matrix (OIM) model, that solves these problems, and determines cardinal directions for even complex regions. The model consists of a tiling phase and an interpretation phase. In the tiling phase , a tiling strategy first determines the zones belonging to the nine cardinal directions of each individual region object and then intersects them. The result leads to a bounded grid called objects interaction grid . For each grid cell the information about the region objects that intersect it is stored in an objects interaction matrix . In the subsequent interpretation phase , a well-defined interpretation method is applied to such a matrix and determines the cardinal direction. Spatial example queries illustrate our new cardinal direction concept that is embedded in a spatial extension of SQL and provides user-defined cardinal direction predicates.

Recently, a wide range of applications like hurricane research, fire management, navigation systems, and transportation has shown increasing interest in managing and analyzing space and time-referenced objects, so-called moving objects, that continuously change their positions over time. In the same way as moving objects can change their location over time, the spatial relationships between them can change over time. An important class of spatial relationships are cardinal directions like north and southeast. In spatial databases and GIS, they characterize the relative directional position between static objects in space and are frequently used as selection and join criteria in spatial queries. Transferred to a spatiotemporal context, the simultaneous location change of different moving objects can imply a temporal evolution of their directional relationships, called development. The goal of this paper is to illustrate, explain, and formally define cardinal direction developments between two moving points. Categories and Subject Descriptors:

Reasoning about space has been a considerable field of study both in Artificial Intelligence and in spatial information theory. Many applications benefit from the inference of new knowledge about the spatial relationships between spatial objects on the basis of already available and explicit spatial relationship knowledge that we call spatial (relationship) facts. Hence, the task is to derive new spatial facts from known spatial facts. A considerable amount of work has focused on reasoning about topological relationships (as a special and important subset of spatial relationships) between simple spatial objects like simple regions. There is a common consensus in the GIS and spatial database communities that simple regions are insufficient to model spatial reality and that complex region objects are needed that allow multiple components and holes. Models for topological relationships between complex regions have already been developed. Hence, as the next logical step, the goal of this paper is to develop a reasoning model for them. Further, no reasoning model considers changes of the spatial fact basis stored in a database between consecutive queries. We show that conventional modeling suffers from performance degradation when the database is frequently changing. Our model does not assume any geometric representation model or data structure for the regions. The model is also backward compatible, i.e., it is also applicable to simple regions.

Range query processing is an important technology in spatial databases. Current studies mainly focus on range queries in outdoor space based on the Euclidean or network distances, but rarely any approach is proposed for indoor space. In this paper, we present two network-based approaches to supporting stationary range queries and continuous range queries for indoor space. There are several challenges involved in designing these approaches. First, having no explicit network construction in indoor space makes it difficult to precisely determine the distances between objects. Second, users can walk to anywhere in the building, which makes it hard to predict users' movements for continuous range queries. Our proposed approaches can overcome these challenges and efficiently support range queries in indoor space.

Apart from visualization tasks, three-dimensional (3D) data management features are not or only hardly available in current spatial database systems and Geographic Information Systems (GIS). But the increasing demands from application domains like urban planning, geoscience, and soil engineering call for systems that are capable of storing, retrieving, querying, and manipulating the underlying 3D spatial data. Current 3D data models are tailored to specific applications and simple 3D spatial objects only, and available 3D data structures are restricted to main memory representations; thus they lack the ability of handling general and complex 3D spatial objects in a database context. Available algorithms, especially intersection algorithms for 3D objects, usually require special properties like convexity or monotonicity. Therefore, universal intersection algorithms that are capable of handling general 3D spatial objects are currently unknown. This paper proposes a paradigm called slice representation as a general data representation method for complex 3D spatial data types. In particular, data structures are developed applying the paradigm to point3D, line3D, surface, and volume data types. Two intersection algorithms that involve one argument of type point3D are introduced and show the benefit of the slice representation.

Spatio-temporal databases deal with geometries changing over time. In general, geometries do not only change discretely but continuously; hence we are dealing with moving objects. In the past, a few moving object data models and query languages have been proposed. Each of them supports either historical movements or future movements but not both together. Consequently, queries that start in the past and extend into the future cannot be supported. To model both historical and future movements of an object, two separate concepts with different properties are required, and extra attention is necessary to avoid their conflicts. Furthermore, current definitions of moving objects are too general and vague. It is unclear how a moving object is allowed to move through space and time. For instance, the continuity or discontinuity of motion is not specified. In this paper, we propose a new moving object data model called Balloon model which provides integrated support for both historical and future movements of moving objects. As part of the model, we provide formal definitions of moving objects with respect to their past and future movements. All kinds of queries including past queries, future queries, and queries that start in the past and end in the future are supported in our model.

Topological relationships between objects in space are of great importance in many disciplines. Recently, topological relationships have been defined for complex spatial objects. However, this definition only expresses topological relationships between complex spatial objects as a whole (global view); therefore,topological information between the individual components (local view)that compose the objects is lost. In this paper we propose a novel, hybrid model of topological relationships for composite regions that provides access to the global topological relationships as well as the local topological relationships that exist between the simple regions that are the components of the composite regions involved.

For a long time topological relationships between spatial objects have been a focus of research in a number of disciplines like artificial intelligence, cognitive science, linguistics, robotics, and spatial reasoning. Especially as predicates they support the design of suitable query languages for spatial data retrieval and analysis in spatial databases and geographical information systems (GIS). Unfortunately, they have so far only been defined for and applicable to simplified abstractions of spatial objects like single points, continuous lines, and simple regions. With the introduction of complex spatial data types an issue arises regarding the design, definition, and number of topological relationships operating on these complex types. This article closes this gap and first introduces definitions of general and versatile spatial data types for complex points , complex lines , and complex regions . Based on the well known 9-intersection model, it then determines the complete sets of mutually exclusive topological relationships for all type combinations. Completeness and mutual exclusion are shown by a proof technique called proof-by-constraint-and-drawing . Due to the resulting large numbers of predicates and the difficulty of handling them, the user is provided with the concepts of topological cluster predicates and topological predicate groups , which permit one to reduce the number of predicates to be dealt with in a user-defined and/or application-specific manner.

Moving objects databases managing spatial objects with continuously changing position and extent over time have recently found large interest in the database community. Queries about moving objects become particularly interesting when they ask for temporal changes in the topological relationships between evolving spatial objects. A concept of spatio-temporal predicates has been proposed to describe these relationships. The goal of this paper is to design efficient algorithms for them so that they can be used in spatio-temporal joins and selections. This paper proposes not to design an algorithm for each new predicate individually but to employ a generic algorithmic scheme, which is able to cover present and future predicate definitions.

Whereas earlier work on spatiotemporal databases generally focused on geometries changing in discrete steps, the emerging area of moving objects databases supports geometries changing continuously. Two important abstractions are moving point and moving region, modelling objects for which only the time-dependent position, or also the shape and extent are relevant, respectively. Examples of the first kind of moving entity are all kinds of vehicles, aircraft, people or animals; of the latter hurricanes, forest fires, forest growth or oil spills in the sea. The goal is to develop data models and query languages as well as DBMS implementations supporting such entities, enabling new kinds of database applications. In earlier work we have proposed an approach based on abstract data types. Hence, moving point or moving region are viewed as data types with suitable operations. For example, a moving point might be projected into the plane, yielding a curve, or a moving region be mapped to a function describing the development of its size, yielding a real-valued function. A careful design of a system of types and operations (an algebra) has been presented, emphasizing completeness, closure, consistency and genericity. This design was given at an abstract level, defining, for example, geometries in terms of infinite point sets. In the next step, a discrete model was presented, offering finite representations and data structures for all the types of the abstract model. The present paper provides the final step towards implementation by studying and developing systematically algorithms for (a large subset of) the operations. Some of them are relatively straightforward; others are quite complex. Algorithms are meant to be used in a database context; we also address filtering techniques and practical issues such as large object management or numeric robustness in the context of an ongoing prototype implementation.

Investigates temporal changes of topological relationships and thereby integrates two important research areas: first, 2D topological relationships that have been investigated quite intensively, and second, the change of spatial information over time. We investigate spatio-temporal predicates, which describe developments of well-known spatial topological relationships. A framework is developed in which spatio-temporal predicates can be obtained by temporal aggregation of elementary spatial predicates and sequential composition. We compare our framework with two other possible approaches: one is based on the observation that spatio-temporal objects correspond to 3D spatial objects for which existing topological predicates can be exploited. The other approach is to consider possible transitions between spatial configurations. These considerations help to identify a canonical set of spatio-temporal predicates.

Spatio-temporal databases deal with geometries changing over time. The goal of our work is to provide a DBMS data model and query language capable of handling such time-dependent geometries, including those changing continuously that describe moving objects . Two fundamental abstractions are moving point and moving region , describing objects for which only the time-dependent position, or position and extent, respectively, are of interest. We propose to present such time-dependent geometries as attribute data types with suitable operations, that is, to provide an abstract data type extension to a DBMS data model and query language. This paper presents a design of such a system of abstract data types. It turns out that besides the main types of interest, moving point and moving region, a relatively large number of auxiliary data types are needed. For example, one needs a line type to represent the projection of a moving point into the plane, or a “moving real” to represent the time-dependent distance of two points. It then becomes crucial to achieve (i) orthogonality in the design of the system, i.e., type constructors can be applied unifomly; (ii) genericity and consistency of operations, i.e., operations range over as many types as possible and behave consistently; and (iii) closure and consistency between structure and operations of nontemporal and related temporal types. Satisfying these goal leads to a simple and expressive system of abstract data types that may be integrated into a query language to yield a powerful language for querying spatio-temporal data, including moving objects. The paper formally defines the types and operations, offers detailed insight into the considerations that went into the design, and exemplifies the use of the abstract data types using SQL. The paper offers a precise and conceptually clean foundation for implementing a spatio-temporal DBMS extension.