8 Using Qualitative Information in Query Proc. over Multiresolution Maps
179
8.6 Consistency of Spatial Relations
When two maps have a set of features in common, possibly represented with different
dimensions, the problem arises of establishing whether such maps represent common
objects in a consistent way. In this chapter we consider topological and cardinal
consistency, that is, consistency with respect to topological and cardinal directional
relationships existing between map objects.
Informally, two maps are consistent when, given any pair of common features,
they share the same or similar spatial relationships in both maps. Different types
of consistency can be defined: equality-based (eq-based) consistency and distancebased (dist-based) consistency. Eq-based consistency requires that the same spatial
relationships exist between each pair of common features in two maps.
Definition 8.6 (Eq-based Consistency). Let θ ∈ T REL∪CREL. f 1 and f 2 in map M 1
are eq-based consistent with f 1 and f 2 in map M 2 , if f 1 θ f 2 holds in both M 1 and M 2 .
M 1 and M 2 are eq-based consistent if, for any pair of features ( f 1 , f 2 ) ∈ (M 1 ∩ M 2 )
2 ,
f 1 and f 2 in map M 1 are eq-based consistent with f 1 and f 2 in map M 2 .
Since topological relations are not defined for all possible pairs of dimensions
(see Table 8.1), eq-based consistency cannot always be guaranteed. For example,
Overlap is defined between pairs of regions and pairs of lines, but it is not defined
between a line and a region. A similar situation arises for multi-tile relations, which
are not defined when the target object is a point.
From this consideration it follows that spatial relationship equality is a too strong
criterion for defining consistency. Such criteria can, however, be relaxed by considering a distance between relations. The new notion of consistency, that we call distbased consistency, is always defined and requires that spatial relationships between
two features in two different maps are not necessarily equal but the most similar ones.
Dist-based consistency seems the most reasonable choice in real situations dealing
with multiresolution, where eq-based consistency cannot always be guaranteed.
In order to formally define dist-based consistency, we rely on the distance functions we have defined in Sects. 8.4.2 and 8.5.2. More precisely, dist-based consistency can be defined by requiring that spatial relationships between pairs of features
in two distinct maps must be the most similar ones, according to the introduced distance functions.
Definition 8.7 (Dist-based Consistency). Let θ 1 , θ 2 ∈ R, where R is either T REL or
CREL. Let f 1 and f 2 be two features appearing in map M 1 with dimensions (d 1 , d 2 )
and in map M 2 with dimensions (d 3 , d 4 ). f 1 and f 2 in map M 1 are dist-based consistent with f 1 and f 2 in map M 2 if f 1 θ 1 f 2 holds in M 1 , f 1 θ 2 f 2 holds in M 2 , and
d h (θ 1 , (d 1 , d 2 ), θ 2 , (d 3 , d 4 )) coincides either with
min{d h (θ 1 , (d 1 , d 2 ), θ 3 , (d 3 , d 4 ))|θ 3 ∈ R(d 3 , d 4 )} or
min{d h (θ 2 , (d 3 , d 4 ), θ 3 , (d 1 , d 2 ))|θ 3 ∈ R(d 1 , d 2 )}
where h ∈ {t, c}, depending on R. M 1 and M 2 are dist-based consistent if for any pair
of features ( f 1 , f 2 ) ∈ (M 1 ∩ M 2 )
2 , f 1 and f 2 in map M 1 are dist-based consistent with
f 1 and f 2 in map M 2 .
179
8.6 Consistency of Spatial Relations
When two maps have a set of features in common, possibly represented with different
dimensions, the problem arises of establishing whether such maps represent common
objects in a consistent way. In this chapter we consider topological and cardinal
consistency, that is, consistency with respect to topological and cardinal directional
relationships existing between map objects.
Informally, two maps are consistent when, given any pair of common features,
they share the same or similar spatial relationships in both maps. Different types
of consistency can be defined: equality-based (eq-based) consistency and distancebased (dist-based) consistency. Eq-based consistency requires that the same spatial
relationships exist between each pair of common features in two maps.
Definition 8.6 (Eq-based Consistency). Let θ ∈ T REL∪CREL. f 1 and f 2 in map M 1
are eq-based consistent with f 1 and f 2 in map M 2 , if f 1 θ f 2 holds in both M 1 and M 2 .
M 1 and M 2 are eq-based consistent if, for any pair of features ( f 1 , f 2 ) ∈ (M 1 ∩ M 2 )
2 ,
f 1 and f 2 in map M 1 are eq-based consistent with f 1 and f 2 in map M 2 .
Since topological relations are not defined for all possible pairs of dimensions
(see Table 8.1), eq-based consistency cannot always be guaranteed. For example,
Overlap is defined between pairs of regions and pairs of lines, but it is not defined
between a line and a region. A similar situation arises for multi-tile relations, which
are not defined when the target object is a point.
From this consideration it follows that spatial relationship equality is a too strong
criterion for defining consistency. Such criteria can, however, be relaxed by considering a distance between relations. The new notion of consistency, that we call distbased consistency, is always defined and requires that spatial relationships between
two features in two different maps are not necessarily equal but the most similar ones.
Dist-based consistency seems the most reasonable choice in real situations dealing
with multiresolution, where eq-based consistency cannot always be guaranteed.
In order to formally define dist-based consistency, we rely on the distance functions we have defined in Sects. 8.4.2 and 8.5.2. More precisely, dist-based consistency can be defined by requiring that spatial relationships between pairs of features
in two distinct maps must be the most similar ones, according to the introduced distance functions.
Definition 8.7 (Dist-based Consistency). Let θ 1 , θ 2 ∈ R, where R is either T REL or
CREL. Let f 1 and f 2 be two features appearing in map M 1 with dimensions (d 1 , d 2 )
and in map M 2 with dimensions (d 3 , d 4 ). f 1 and f 2 in map M 1 are dist-based consistent with f 1 and f 2 in map M 2 if f 1 θ 1 f 2 holds in M 1 , f 1 θ 2 f 2 holds in M 2 , and
d h (θ 1 , (d 1 , d 2 ), θ 2 , (d 3 , d 4 )) coincides either with
min{d h (θ 1 , (d 1 , d 2 ), θ 3 , (d 3 , d 4 ))|θ 3 ∈ R(d 3 , d 4 )} or
min{d h (θ 2 , (d 3 , d 4 ), θ 3 , (d 1 , d 2 ))|θ 3 ∈ R(d 1 , d 2 )}
where h ∈ {t, c}, depending on R. M 1 and M 2 are dist-based consistent if for any pair
of features ( f 1 , f 2 ) ∈ (M 1 ∩ M 2 )
2 , f 1 and f 2 in map M 1 are dist-based consistent with
f 1 and f 2 in map M 2 .