6 Automated Geographical Information Fusion
117
In plain language, the rule above states that if the extensions of two categories x and y
are disjoint then we infer that the categories themselves are incomparable. A further
obvious rule, suggested by the category Forest & Urban in Fig. 6.1, is to create a
new category corresponding to two overlapping category extents as follows:
for all x ∈ C 1 and y ∈ C 2 if e 1 (x) ∩ e 2 (y) ∅
and e 1 (x) e 2 (y) and e 2 (x) e 1 (y)
then x ∩ y ∈ C f and x ∩ y ≤ f x and x ∩ y ≤ f y
The rule above creates a new category x ∩ y in the fused taxonomy that lies at the
intersection of categories x and y. For two data sets G 1 and G 2 , the conclusions from
such rules form an ordering relation that relates categories in the two source taxonomies. Together with those source-ordering relations, this enables the derivation
of a new fused partial order in G f that defines the subsumption relationships between
categories within the different taxonomies of G 1 and G 2 .
Once formalized, these rules can be implemented within an automated reasoning
system. Indeed, early versions of our rosetta system adopted this approach, using
the RACER description logic engine [35] for automated reasoning. An advantage of
using description logics for this purpose is that any inconsistencies between the chosen rules can be automatically detected using the consistency and satisfiability services provided by any description logic. Having generated the ontology alignment,
the spatial data itself can then be automatically fused based on the standard geographical information integration techniques (i.e. overlay the two spatial data sets,
and assign to each fused region the category in the fused partial order that lies at the
intersection of the two source categories for the fused region).
Algebraic System
The reasoning system approach described above provides an important step on the
road to practical automated geographical information fusion systems. However, it
has at least two important shortcomings.
First, a partial order is a rather too general structure for describing a geographical
ontology. For each pair of input categories we need to be able to identify a unique
category in our fused taxonomy that corresponds to the fusion of those input categories. Using partial orders, it may not be possible to guarantee that such a unique
fused category exists, since a pair of elements in a partial order may have multiple
incomparable least upper and greatest lower bounds. A more appropriate structure is
a lattice, which as we have already seen is commonly used in formal approaches to
ontological information [29, 37, 60]. A lattice is a special type of partial order, where
all subsets of elements have a unique least upper bound and a unique greatest lower
bound in the lattice. The simplified taxonomies in Fig. 6.1 and subsequent figures
can be represented as lattices.
1
1 Strictly, the taxonomies in the figures in this chapter are shown as join semi-lattices, but
any finite join semi-lattice can be trivially transformed into a lattice with the addition of a
bottom element.
117
In plain language, the rule above states that if the extensions of two categories x and y
are disjoint then we infer that the categories themselves are incomparable. A further
obvious rule, suggested by the category Forest & Urban in Fig. 6.1, is to create a
new category corresponding to two overlapping category extents as follows:
for all x ∈ C 1 and y ∈ C 2 if e 1 (x) ∩ e 2 (y) ∅
and e 1 (x) e 2 (y) and e 2 (x) e 1 (y)
then x ∩ y ∈ C f and x ∩ y ≤ f x and x ∩ y ≤ f y
The rule above creates a new category x ∩ y in the fused taxonomy that lies at the
intersection of categories x and y. For two data sets G 1 and G 2 , the conclusions from
such rules form an ordering relation that relates categories in the two source taxonomies. Together with those source-ordering relations, this enables the derivation
of a new fused partial order in G f that defines the subsumption relationships between
categories within the different taxonomies of G 1 and G 2 .
Once formalized, these rules can be implemented within an automated reasoning
system. Indeed, early versions of our rosetta system adopted this approach, using
the RACER description logic engine [35] for automated reasoning. An advantage of
using description logics for this purpose is that any inconsistencies between the chosen rules can be automatically detected using the consistency and satisfiability services provided by any description logic. Having generated the ontology alignment,
the spatial data itself can then be automatically fused based on the standard geographical information integration techniques (i.e. overlay the two spatial data sets,
and assign to each fused region the category in the fused partial order that lies at the
intersection of the two source categories for the fused region).
Algebraic System
The reasoning system approach described above provides an important step on the
road to practical automated geographical information fusion systems. However, it
has at least two important shortcomings.
First, a partial order is a rather too general structure for describing a geographical
ontology. For each pair of input categories we need to be able to identify a unique
category in our fused taxonomy that corresponds to the fusion of those input categories. Using partial orders, it may not be possible to guarantee that such a unique
fused category exists, since a pair of elements in a partial order may have multiple
incomparable least upper and greatest lower bounds. A more appropriate structure is
a lattice, which as we have already seen is commonly used in formal approaches to
ontological information [29, 37, 60]. A lattice is a special type of partial order, where
all subsets of elements have a unique least upper bound and a unique greatest lower
bound in the lattice. The simplified taxonomies in Fig. 6.1 and subsequent figures
can be represented as lattices.
1
1 Strictly, the taxonomies in the figures in this chapter are shown as join semi-lattices, but
any finite join semi-lattice can be trivially transformed into a lattice with the addition of a
bottom element.