Egenhofer, M.J., Clementini, E. and Di Felice, P., Topological relations between regions with holes, International Journal of Geographical Information Systems, vol 8, no 2, pp 129---142, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....2.1.13.1 Background The Relational Operators are Boolean methods that are used to test for the existence of a specified topological spatial relationship between two geometries. Topological spatial relationships between two geometric objects have been a topic of extensive study in the literature [4,5,6,7,8,9,10]. The basic approach to comparing two geometries is to make pair wise tests of the intersections between the Interiors, Boundaries and Exteriors of the two geometries and to classify the relationship between the two geometries based on the entries in the resulting intersection matrix. The ....

Egenhofer, M.J., Clementini, E. and Di Felice, P., Topological relations between regions with holes, International Journal of Geographical Information Systems, vol 8, no 2, pp 129---142, 1994.

....regions with multiple parts and possibly with holes. It is surprising that topological predicates on crisp complex regions have so far not been defined. In [3] the so called TRCR (Topological Relationships for Composite Regions) model only allows sets of disjoint simple regions without holes. In [7] only topological relationships of simple regions with holes are considered. Topological predicates on fuzzy spatial objects, let them be simple or complex, have so far not been defined. 3 A Model for Crisp and Fuzzy Complex Regions In this paper we only consider topological predicates that ....

M.J. Egenhofer, E. Clementini, and P. Di Felice. Topological Relations between Regions with Holes. Int. Journal of Geographical Information Systems, 8(2):128-- 142, 1994.

....allows sets of disjoint simple regions without holes. But topological relationships between composite regions are defined in an ad hoc manner and are not systematically derived from the underlying model. Moreover, the model is only related to but not directly based on the 9 intersection model. In [6] topological relationships of simple regions with holes are considered. Unfortunately, multi part regions are not permitted. While the authors take the number of components (area without holes, holes) of two regions into account and consider the large number of topological relationships between ....

M.J. Egenhofer, E. Clementini, and P. Di Felice. Topological Relations between Regions with Holes. Int. Journal of Geographical Information Systems, 8(2):128-- 142, 1994.

....have been adopted to include not only the dimension of the resulting components but also the number of occurrences of particular intersections. The approach has been applied using regions and has been extended for the representation of relationships between lines [11] and regions with holes [15]. Egenhofer s approach and the various extensions thereof have been 2 Egenhofer [9] discusses elimination of the impossible cases. 3 used to represent topological relationships only. For this purpose the approach has proved to be complete but not sound. In order to achieve soundness additional ....

Egenhofer, M.J., Clementini, E., Di Felice P. (1994) Topological Relations Between Regions With Holes, Int. J. Geographic Information Systems, 8, 2, 129-142

....A, and are enclosed by A. Unfortunately, ordinary point set topology offers no method to extract holes from a (regular closed) point set as separate components; they are simply part of the complement. Note that this does not mean that regions with holes cannot be modeled. Some research work in [ECF94, Sch97, WB93], for example, shows that this is possible by selecting a constructive approach. Roughly speaking, the idea is to assume that the holes of A are already given as regions and to subtract these holes from a generalized region A being isomorphic to a closed disc and being the union of A and the ....

M.J. Egenhofer, E. Clementini, and P. di Felice. Topological Relations between Regions with Holes. Int. Journal of Geographical Information Systems, 8(2):128--142, 1994.

....of the additional sets the exterior of the treated regions is much larger, or goes to infinity, compared to the size of the treated regions themselves Egenhofer and Herring propose a combination of the 4 intersection and additional criterion to resolve ambiguities in the extended analyses. Egenhofer et al. 1994a) who are also interested on regions with holes, solve ambiguities by reasoning about generalized regions, neglecting the holes. But as we will see, the complexity of calculation with the exterior of spatial objects can be eliminated (Sect. 4.4) In this paper we confine ourselves to regions ....

....2Dg Egenhofer (1993) also specifies topological relations in a more detailed way, by additional numerical topological invariants: the dimension of intersections, and the number of components of an intersection set. Other topological invariants are added in a multiple representation framework in Egenhofer et al. 1994b) 4 The Hyper Raster Model Now we propose to applicate the cellular decomposition of IR 2 (cf. Sect. 2.3) as a data model for digital images. We develop here a data structure for the hyper raster, which we will use later for topological reasoning (Sect. 4.4) A sketch of the hyper raster ....

Egenhofer, Max J.; Clementini, Eliseo; di Felice, Paolino (1994): Topological relations between regions with holes. International Journal of Geographical Information Systems, 8(2):129--142, 1994.

....analysis of the the data need be performed. Often the analysis of topological relations may reduce the burden of geometric computations. A formal analysis of relations between sets has been provided by Egenhofer in several publications (Egenhofer and Franzosa 1991, Egenhofer 1991, Egenhofer 1993, Egenhofer et al. 1994). The idea is to represent the mutual relations of two sets A and B by a 2 by 2 matrix, called the 4intersection F, which is given by F = A B A B ffi A ffi B A ffi B ffi (1) where A; B denote the boundaries, A ffi ; B ffi the interiors of the regions A and B, and ....

Egenhofer, Max J.; Clementini, Eliseo; di Felice, Paolino (1994): Topological relations between regions with holes. International Journal of Geographical Information Systems, 8(2):129--142, 1994.

....they are composed of several levels, arranged into a hierarchy. Each level has its own semantics, although it can also be seen as a refinement of the preceding level. HPS was designed to represent such hierarchical models in a consistent way. Thus, HPS addresses a concern expressed by Egenhofer [19]: a major impediment in the transition to more powerful multiple representation GIS is the lack of methods to maintain consistently the multiple representation of geographic objects . Another common use of hierarchy is in the context of multi resolution maps [20, 21] In this case, the goal is to ....

M. J. Egenhofer, E. Clementini, and P. Di Felice. Topological relations between regions with holes. Int. Journal of Geographical Information Systems, 8(2):129--142, 1994.

....model that offers explicit description of spatial objects, and efficient encoding retrieval of spatial relations. A fair amount of work has been done in the last few years in the direction of a formal approach to the description and manipulation of spatial entities and their relationships [8, 17, 9, 27, 16, 5, 28, 12], and a number of models have been proposed in the literature for giving a comprehensive representation of the geometric structure of plane geographic maps (see, e.g. 14, 7, 21, 27, 22, 5] Different models are characterised mostly by their expressive power, defined by the degree of generality ....

....overlap, inside, and coveredBy. For regions with holes the classification is obtained by combining the 4 intersections among the simply connected regions interior to the outer boundaries (i.e. those obtained by eliminating the holes) and the simply connected regions corresponding to the holes [12]. Topological relations can be extended easily to pairs of atomic entities, including also lines and points: in this case, also relations bounds and boundedBy are possible, between entities of different dimensions, and such that one is contained in the boundary of the other [5] A map is often ....

Egenhofer, M., Clementini, E., Di Felice, P.: Topological relations between regions with holes. International Journal of Geographical Information Systems, 8 (2), 1994, pp.129-142

....designing efficient algorithms and data structures. These efforts give an evidence that there is an obvious need of a formal model for the representation of topological relationships. Recently there has been a growing interests in the computer science and GIS community in topological relationships [1, 5, 8, 10, 15] and topological queries [2, 4, 20 22] A large number of contributions in the literature have focused on the modeling of topological relationships between simple objects: simply connected homogeneous 2 dimensional areas, simple lines (i.e. piecewise algebraic curves with exactly two endpoints) ....

....of these topological relationships holds between any two homogeneous 2 dimensional point sets in IR 2 (see Fig. 1) In a variant of this model, called nine intersection model, intersections with the exteriors are added, leading to a classification of binary topological relationships as a 9 tuple [6, 8]. 1 The model for describing the binary topological relationships between simple regions has been generalized nicely to other geometric object types: line line, line region, point region, etc. 3] The original four intersection model has recently been also extended to more complex types of ....

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M. J. Egenhofer, E. Clementini, and P. Di Felice. Topological relations between regions with holes. Int. J. Geographical Inf. Syst., 8(2):129--142, 1994.

....also exist, they are beyond the scope of this chapter. Spatial relationships are defined on pairs of spatial entities, and depend on their relative positions in space. There is a flourishing literature on the definition, classification, evaluation, and study of invariants of spatial relationships [Cle95, Cle96a, DeF93, Ege91, Ege94a, Had92, Jor96, Kai90, Wor92, Wor93]. In spite of all such work, a homogeneous and complete characterization (e.g. based on an algebraic approach) is still missing. It is not uncommon to find practical contexts and applications for which none of the models proposed in the literature is satisfactory. However, spatial relations are ....

Egenhofer, M., Clementini, E., Di Felice, P., Topological relations between regions with holes, International Journal of Geographical Information Systems, 8, 2, 1994, pp.129-142

....of qualitative relations it is, for our knowledge, a completely new question. The presented ideas are worked out for simple regions and conditions for a small positional uncertainty. Further work should extend these ideas for complex objects an intermediate step could be generalized regions (Egenhofer et al. 1994) , or for objects of other dimensions. It is to proof that the ideas hold for simple 1D objects in IR, or for simple 3D objects in IR 3 . Also it is to investigate whether the method is to transfer to other qualitative spatial relationships. Another aspect of further research is the extension ....

Egenhofer, Max J.; Clementini, Eliseo; di Felice, Paolino (1994): Topological relations between regions with holes. International Journal of Geographical Information Systems, 8(2):129--142, 1994.

....in particular ordering, and metrical relations interact. Besides this, we are planning to do further empirical investigations of the same kind where the regions we present are not restricted to circles. Apart from allowing arbitrary shape, it seems to be very interesting to use regions with holes (Egenhofer, Clementini Di Felice, 1994) as relations between these regions can be expressed using RCC 8, although the assignment of these relations to pairs of regions doesn t seem to be straightforward. Last but not least we are planning to conduct experiments on the inferential adequacy of the RCC theory, as we have done previously ....

Egenhofer, M. J., Clementini, E. & Di Felice, P (1994). Topological relations between regions with holes. Int. Journal of Geographical Information Systems, 2, 129-144.

....regions. We will introduce this model only informally here. A formal definition of this model based on the point set paradigm and on point set topology is given in the Appendix. Each alternative model should fulfill the properties described there. Possible candidates are the models described in [ECF94, WB93], and the discrete model of the ROSE algebra [GS93, GS95, Sc95] A (determinate) region is a set of disjoint, connected areal components possibly with disjoint holes (see the picture below) This model is very general and closed under (appropriately defined) geometric union, intersection, ....

M.J. Egenhofer, E. Clementini & P. di Felice. Topological Relations between Regions with Holes. Int. Journal of Geographical Information Systems, vol. 8, no. 2, pp. 129-142, 1994.

....relations. To enable more comprehensive cyclic reasoning it is necessary to establish the composition of the sixteen cyclic relations (e.g. A meets twice B and B contained by C implies A overlaps twice C) Based on a method used for determining the composition of topological relations in IR 2 [24], we will derive all 256 compositions for the cyclic relations. Of particular interest will be the crispness of these compositions as compared to the crispness of the compositions for linear intervals [18] Further extensions to the model are also possible, for example, future work will include ....

Egenhofer, M., E. Clementini, and P. Felice, Topological relations between regions with holes. International Journal of Geographical Information Systems, 1994. 8(2): p. 129142.

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