Abstract
An abstract examination of refinement (and conversely, coarsening) with respect to the involved spatial relations gives rise to formulated order relations between spatial coverings, which are defined as complete-coverage representations composed of regional granules. Coverings, which generalize partitions by allowing granules to overlap, enhance hierarchical geocomputations in several ways. Refinement between spatial coverings has underlying patterns with respectto inclusion—formalized as binary topological relations— between their granules. The patterns are captured by collection relations of inclusion, which are obtained by constraining relevant topological relations with cardinality prop- erties such as uniqueness and totality. Conjoining relevant collection relations of equality and proper inclusion with the overlappedness (non-overlapped or overlapped) of the re- fining and the refined covering yields collection order relations, which serve as specific types of refinement between spatial coverings. The examination results in 75 collection order relations including seven types of equality and 34 pairs of strict or non-strict types of refinement and coarsening, out of which 19 pairs form partial collection orders.
Highlights
Refinement [16] plays an important role in analyzing and reasoning with spatial hierarchies
This paper develops a formal model of spatial refinement between coverings in the following four steps
This paper examined spatial refinement between coverings, which differentiate by overlappedness into partitions and o-coverings, and thereby offer enhanced ways of dealing with multiple spatial datasets and modeling real-world geographic features
Summary
Refinement (and coarsening) [16] plays an important role in analyzing and reasoning with spatial hierarchies. The hierarchical nature of spatial knowledge [17] has been studied via multiple approaches and modeled with various spatial data structures. An abstract examination of refinement between complete-coverage representations, such as partitions and coverings, reveals the underlying patterns of the involved spatial rec by the author(s). ZHAO lations, which leads to formal interpretations of spatial refinement with a series of order relations, and thereby potentially offers new insights into hierarchical geocomputations
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