A system of types and operators for handling vague spatial objects

Abstract

Vagueness is often present in spatial phenomena. Representing and analysing vague spatial phenomena requires vague objects and operators, whereas current GIS and spatial databases can only handle crisp objects. This paper provides mathematical definitions for vague object types and operators.The object types that we propose are a set of simple types, a set of general types, and vague partitions. The simple types represent identifiable objects of a simple structure, i.e. not divisible into components. They are vague points, vague lines, and vague regions. The general types represent classes of simple type objects. They are vague multipoint, vague multiline, and vague multiregion. General types assure closure under set operators. Simple and general types are defined as fuzzy sets in 2 satisfying specific properties that are expressed in terms of topological notions. These properties assure that set membership values change mostly gradually, allowing stepwise jumps. The type vague partition is a collection of vague multiregions that might intersect each other only at their transition boundaries. It allows for a soft classification of space. All types allow for both a finite and an infinite number of transition levels. They include crisp objects as special cases.We consider a standard set of operators on crisp objects and define them for vague objects. We provide definitions for operators returning spatial types. They are regularized fuzzy set operators: union, intersection, and difference; two operators from topology: boundary and frontier; and two operators on vague partitions: overlay and fusion. Other spatial operators, topological predicates and metric operators, are introduced giving their intuition and example definitions. All these operators include crisp operators as special cases. Types and operators provided in this paper form a model for a spatial data system that can handle vague information. The paper is illustrated with an application of vague objects in coastal erosion.

Plateau Regions: An Implementation Concept for Fuzzy Regions in Spatial Databases and GIS

“…These concepts have been deliberately developed as a specification for a possible implementation in spatial database systems. A similar type system of so-called vague spatial data types is introduced in [5]. Instead of the term "fuzzy", the authors use the term "vague".…”

“…We model such a spatial phenomenon by a fuzzy region. Fuzzy set theory [10] has been a popular approach to modeling vague spatial objects and resulted in a concept of fuzzy regions [5,8]. A crisp region object is conceptually modeled as a particular point set of the Euclidean plane [7,9].…”

Querying Vague Spatial Objects in Databases with VASA

“…Approaches in this category include earlier fuzzy regions [3], the formal definition of fuzzy points, fuzzy lines and fuzzy regions in [23], and an extension of the rough classification from [1] to account for fuzzy regions [2]. A recent effort for the definition of a spatial algebra based on fuzzy sets is presented in [9]. Finally, probabilistic approaches [13] focus on an expected membership to an object which can be contrasted to the membership values of fuzzy sets that are objective in the sense that they can be computed formally or determined empirically.…”

Soft Computing Techniques in Spatial Databases

“…However, there is an increasing interest in applying fuzzy theory to spatial topics. Examples are given by Altman (1994) presenting fuzzy set theoretic approaches for handling imprecision in spatial analysis and introducing fuzzy regions in ℕ 2 , Schneider (1999) providing the foundation of FUSA in terms of fuzzy spatial data types for fuzzy points, fuzzy lines, and fuzzy regions together with some fuzzy spatial operations, Dilo, de By & Stein (2007) introducing a similar type system of fuzzy (called "vague") spatial data types with a more comprehensive set of operations, Schneider (2003) proposing a conceptual model and an implementation model of fuzzy spatial objects that are not defined on the Euclidean plane but on a discrete geometric domain called grid partition, and Schneider (2000) presenting metric operations on fuzzy spatial objects like the area of a fuzzy region or the length of a fuzzy line. Different models of fuzzy topological predicates, which are not discussed in this chapter and which characterize the relative position of two fuzzy spatial objects towards each other, are discussed in Petry et al (2002), Schneider (2001aSchneider ( , 2001b, Shi & Guo (1999), Tang & Kainz (2002), and Zhan (1997.…”