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 ℝ2 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.