A system of types and operators for handling vague spatial objects

models for spatial data propose a list of types and fundamental operators that are needed in spatial data systems. The ROSE algebra (Güting 1994, Güting and Schneider 1995) and OpenGIS abstract specification (Open GIS Consortium 1999) both provide a similar list of spatial operators. (A comparison between the two models can be found in Dilo (2006).) To present the list of operators, we follow here the grouping provided in Güting (1994). (i) Operators returning spatial data type values, e.g. intersection of regions— returning regions, union of lines—returning lines, boundary of regions— returning lines. (ii) Spatial predicates expressing spatial relations, e.g. point within region, region overlaps region. (iii) Spatial operators returning numbers. These are mainly metric operators e.g. length of a line, area of a region, distance between two regions. (iv) Operators on collection of objects, e.g. overlay of partitions, fusion of regions based on the equality of values of a certain attribute. Vague spatial types introduced in section 5 form a complete set of types describing spatial objects. Some numerical types are needed for the other operators, spatial predicates and metric operators. The type TruthDegree;[0, 1] is the return type of all spatial predicates. Two other types, Measure; + , and VMeasure;{f:(0, s]R + |s g (0, 1] and f is semi-continuous} are needed for metric operators. The vague operators introduced in section 5 cover the first and fourth group of spatial operators: operators returning spatial data types and operators on collections. Other operators returning spatial types can be expressed as a combination of union, intersection and complement, e.g. difference and symmetric difference. In this section we give intuition and example definitions for operators of the two other groups, spatial predicates and metric operators. Spatial predicates express relations between vague multipoints, vague multilines and vague multiregions. We provide relations Disjoint, Touches, Crosses, Overlaps, Within, and Equal, which is the set of relations proposed by SQL/MM spatial (ISO 1999). The relations extend the true/false set of truth values of the SQL/MM relations to the [0, 1] interval. That means that the truth of a relation is a matter of degree, thus a value of type TruthDegree. A value v between 0 and 1 for a relation R(m, n) means that objects m and n are in relation R to the degree v. A value 0 for R(m, n) means that m and n are certainly not in relation R, whereas a value 1 means the two are certainly in R. Spatial relations are defined from membership values of the objects involved, considering extreme values that support a relation or disapprove it. The complete treatment of spatial relations, intuition, definition and illustrations can be found in Dilo et al. (2005) and Dilo (2006). Here we give the properties of the relations, and some example definitions. The relations Disjoint, Touches, Crosses and Overlaps between vague objects are defined such that a relation is certain if the corresponding crisp relation is true for their cores; a relation is certainly false if the corresponding 1 More operators are proposed in the ROSE algebra from this last group. 418 A. Dilo et al. D o w n l o a d e d B y : [ U n i v e r s i t y o f C o l o r a d o , B o u l d e r c a m p u s ] A t : 1 3 : 0 9 1 8 J u l y 2 0 0 9 crisp relation is false for their support sets. For example, the relation Disjoint is defined as: Disjoint : GVSpatial|GVSpatial?TruthDegree Vm, n[GVSpatial, Disjoint m, n ð Þ~1{supp[ 2 myn ð Þ p ð Þ f g: The total certainty of the other two relations, Within and Equal, is modelled by the subset and equality relation for fuzzy sets, respectively. A Within(m, n) relation is certainly false if the corresponding crisp relation between the core of m and the support set of n is false. Similarly, an Equal relation is certainly false if the corresponding crisp relation between the core of one object and the support set of the other is false. The relation Within is defined from the bounded difference between two fuzzy sets m and n: ;p, m,n(p)5max{0, m(p)2n(p)}. The Within relation is defined as: Within : GVSpatial|GVSpatial?TruthDegree Vm, n[GVSpatial, Within m, n ð Þ~ 0 if myn~0 2 , 1{supp m+n ð Þ p ð Þ f g otherwise: ( The relations have the property that only one relation can be certain at a time, i.e. if one relation is certain, all the others have a degree smaller than 1. For some of the relations this property is stronger: if a relation is certain, all the others are false. Each relation gives the corresponding crisp relation when applied to crisp objects. Metric operators that we provide are distance between two vague objects of any type, length of a vague multiline, area, diameter and perimeter of a vague multiregion. An operator on vague objects is such that for every a in (0, 1] it returns the value of the analogous crisp operator applied to the a-cuts of the vague objects. For example, the area operator returns for every a in (0, 1] the area of the a-cut of the vague multiregion. We call these alpha operators. An alpha operator takes as argument one or two vague objects, and returns a function from an interval in (0, 1] to the non-negative real numbers + . The returned function by an alpha operator is a value of type VMeasure. Again, we give here only example definitions. The complete treatment of metric operators can be found in Dilo (2006). The alpha area of a vague multiregion m is calculated from the areas of its a-cuts: AREA ma ð Þ~ Ð Ð ma dx dy. If the maximum of m is lower than 1, we consider the area Area(m) to be 0 for all a values higher than the maximum. The Area operator is defined as: Area : VMRegion?VMeasure Vm[VMRegion, Area m 1⁄2 a ð Þ~ Area ma ð Þ 0vaƒmaxp m p ð Þ f g, 0 maxp m p ð Þ f gvaƒ1: ( For each alpha operator we provide a corresponding operator that produces an average over all values of the return function of the alpha operator. We call the operators of this second group average operators. As the integration performs an averaging process on functions, we define an average operator as the integral over A system of types and operators for handling vague spatial objects 419 D o w n l o a d e d B y : [ U n i v e r s i t y o f C o l o r a d o , B o u l d e r c a m p u s ] A t : 1 3 : 0 9 1 8 J u l y 2 0 0 9 [0, 1] of the return function of the corresponding alpha operator. An average operator returns a non-negative real number that is a value of type Measure. As an example operator from this group see the average area operator. The alpha area of a vague multiregion is a non-increasing and upper semicontinuous function, thus the function Area(m) is integrable. The average area of a region m is calculated from the integral of Area(m) over [0, 1]. The operator AvArea is defined as: AvArea : VMRegion?Measure Vm[VMRegion, AvArea m ð Þ~ ð1 0 Area m 1⁄2 a ð Þda: The average area calculated from the integral Area[m](a)da provides the volume under the m function. Therefore, the average area is equal to