Spatial database systems and geographical information systems are currently only able to support geographical applications that deal with crisp spatial objects, that is, objects whose extent, shape, and boundary are precisely determined. Examples are land parcels, school districts, and state territories. However, many new, emerging applications are interested in modeling and processing geographic data that are inherently characterized by spatial vagueness or spatial indeterminacy. This requires novel concepts due to the lack of adequate approaches and systems. In this chapter, we focus on an important kind of spatial vagueness called spatial fuzziness. Spatial fuzziness captures the property of many spatial objects in reality that do not have sharp boundaries and interiors or whose boundaries and interiors cannot be precisely determined. We will designate this kind of entities as fuzzy spatial objects. Examples are polluted areas, temperature zones, and lakes. We propose an abstract, formal, and conceptual model of so-called fuzzy spatial data types (that is, a fuzzy spatial algebra) introducing fuzzy points, fuzzy lines, and fuzzy regions in the two-dimensional Euclidean space. This chapter provides a definition of their structure and semantics, which is supposed to serve as a specification of their implementation. Furthermore, we introduce fuzzy spatial set operations like fuzzy union, fuzzy intersection, and fuzzy difference, as well as fuzzy topological predicates as they are useful in fuzzy spatial joins and fuzzy spatial selections. We also sketch implementation strategies for the whole type system and show their integration into databases. An outlook on future research challenges rounds out the chapter.
Key Terms in this Chapter
Fuzzy Spatial Algebra: It is a system of fuzzy spatial data types including a comprehensive set of fuzzy spatial operations and fuzzy spatial predicates and satisfying closure properties.
Geometric Anomaly: This occurs when the results of geometric set operations on fuzzy regions are, from an application standpoint, considered degeneracies like isolated or dangling point or line features and missing points and lines in the form of cuts and punctures in the interior of regions.
Regularization: It is a formal concept based on fuzzy topology that removes geometric anomalies on fuzzy regions.
Topological Space: It is a set X together with a collection T of subsets of X satisfying the following axioms. (a) The empty set and X are in T. (b) The union of any collection of sets in T is also in T. (c) The intersection of any pair of sets in T is also in T. The collection T is called a topology on X, and the elements of X are called points. Under this definition, the sets in T are the open sets, and their complements in X are the closed sets. The requirement that the union of any collection of open sets be open is more stringent than simply requiring that all pairwise unions be open as the former includes unions of infinite collections of sets.
Fuzzy Spatial Data Type: It is a data type for representing a fuzzy point, fuzzy line, or fuzzy region object that can be employed as an attribute data type in a database system.
Spatial Fuzziness, Spatial Vagueness: Inherent property of many spatial objects in reality that do not have sharp boundaries or whose boundaries cannot be precisely determined.
Fuzzy Topological Predicate: A fuzzy topological predicate characterizes the relative position of two fuzzy spatial objects to each other.
Fuzzy Spatial Query Language: This is a full-fledged query language that integrates fuzzy spatial data types, operations, predicates, modifiers, and other fuzzy concepts.
Spatial Database System: It is a full-fledged database system that, in addition to the functionality of standard database systems for alphanumeric data, provides special support for the storage, retrieval, management, and querying of spatial data, that is, objects in space.