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Paola Podest` a, Barbara Catania, and Alberto Belussi
information coming from different sources, leading to maps with different resolutions. In order to use such data sets in a multiresolution GIS, the problem arises of
determining their consistency.
Consistency checking can be performed by using various consistency measures
and, in case of negative result, pairs of features which more contribute to the inconsistency situation should be identified and ordered according to the distance value.
These pairs could be considered as candidate errors and the list of all inconsistent
objects can be used to generate, by difference, some consistent views of the input
maps, to be consistently used inside GIS applications.
For example, we may be interested in determining whether maps M 1 , M 2 , and M 3
in Fig. 8.4 are consistent. The main problem in checking consistency is due to the fact
that the same object can be represented in distinct maps with different dimensions
(consider, for example, objects T 4 and R 5 in maps M 1 , M 2 , and M 3 ). In order to evaluate consistency between pairs of maps, in Sect. 8.6 we define two distinct notions
of consistency based on the proposed topological and cardinal direction distances.
8.4 Topological Relations
In the following, we first present a formal model for topological relations; then, we
introduce a distance function for comparing two topological relationships, possibly
defined over multiresolution objects.
8.4.1 The Model
In defining a distance function for topological relationships, we consider the following set of topological relationships: T REL = {Dis joint, T ouch, In, Contain, Equal,
Cross, Overlap, Cover, CoveredBy}. Each topological relation corresponds to a set
of 9-intersection matrices (see Sect. 8.2.1), according to the model presented in [6].
We notice that relations Cover and CoveredBy are here defined as refinements of relations Contain and In and are not considered in [6]. The semantics of the topological
predicates of T REL is provided in Table 8.1.
As we can see from Table 8.1, not all relationships can be defined for any pair
of dimensions. Therefore, given two dimensions d 1 , d 2 ∈ {R, L, P}, in the following we denote with T REL(d 1 , d 2 ) the set of topological relationships that can be
defined for d 1 and d 2 . Moreover, we denote with I 9 (θ, d 1 , d 2 ) the set of 9-intersection
matrices defining predicate θ ∈ T REL between two objects of dimension d 1 and d 2 .
8.4.2 Distance Function for Topological Relationships
In order to define a distance function for topological relationships in T REL, as each
topological relationship in T REL corresponds to a set of 9-intersection matrices,
we use a two-step approach: first, a distance function between two 9-intersection
matrices is defined then such function is used for computing the final result.
Paola Podest` a, Barbara Catania, and Alberto Belussi
information coming from different sources, leading to maps with different resolutions. In order to use such data sets in a multiresolution GIS, the problem arises of
determining their consistency.
Consistency checking can be performed by using various consistency measures
and, in case of negative result, pairs of features which more contribute to the inconsistency situation should be identified and ordered according to the distance value.
These pairs could be considered as candidate errors and the list of all inconsistent
objects can be used to generate, by difference, some consistent views of the input
maps, to be consistently used inside GIS applications.
For example, we may be interested in determining whether maps M 1 , M 2 , and M 3
in Fig. 8.4 are consistent. The main problem in checking consistency is due to the fact
that the same object can be represented in distinct maps with different dimensions
(consider, for example, objects T 4 and R 5 in maps M 1 , M 2 , and M 3 ). In order to evaluate consistency between pairs of maps, in Sect. 8.6 we define two distinct notions
of consistency based on the proposed topological and cardinal direction distances.
8.4 Topological Relations
In the following, we first present a formal model for topological relations; then, we
introduce a distance function for comparing two topological relationships, possibly
defined over multiresolution objects.
8.4.1 The Model
In defining a distance function for topological relationships, we consider the following set of topological relationships: T REL = {Dis joint, T ouch, In, Contain, Equal,
Cross, Overlap, Cover, CoveredBy}. Each topological relation corresponds to a set
of 9-intersection matrices (see Sect. 8.2.1), according to the model presented in [6].
We notice that relations Cover and CoveredBy are here defined as refinements of relations Contain and In and are not considered in [6]. The semantics of the topological
predicates of T REL is provided in Table 8.1.
As we can see from Table 8.1, not all relationships can be defined for any pair
of dimensions. Therefore, given two dimensions d 1 , d 2 ∈ {R, L, P}, in the following we denote with T REL(d 1 , d 2 ) the set of topological relationships that can be
defined for d 1 and d 2 . Moreover, we denote with I 9 (θ, d 1 , d 2 ) the set of 9-intersection
matrices defining predicate θ ∈ T REL between two objects of dimension d 1 and d 2 .
8.4.2 Distance Function for Topological Relationships
In order to define a distance function for topological relationships in T REL, as each
topological relationship in T REL corresponds to a set of 9-intersection matrices,
we use a two-step approach: first, a distance function between two 9-intersection
matrices is defined then such function is used for computing the final result.