The importance of topological and directional relationships between
spatial objects has been stressed in different fields, notably in
Geographic Information Systems (GIS). In an earlier work, we introduced
the notion of the F-histogram, a generic quantitative representation of
the relative position between two 2D objects, and showed that it can be
of great use in understanding the spatial organization of regions in
images. Here, we illustrate that the F-histogram constitutes a valuable
tool for extracting directional and topological relationship
information. The considered objects are not necessarily convex and
their geometry is not approximated through, e.g., Minimum Bounding
Rectangles (MBRs). The F-histograms introduced in this chapter are
coupled with Allen’s temporal relationships based on fuzzy set theory.
Allen’s relationships are commonly extended into the spatial domain for
GIS purposes, and fuzzy set theoretic approaches are widely used to
handle imprecision and achieve robustness in spatial analysis. For any
direction in the plane, the F-histograms define a fuzzy 13-partition of
the set of all object pairs, and each class of the partition
corresponds to an Allen relation. Lots of directional and topological
relationship information as well as different levels of refinements can
be easily obtained from this approach, in a computationally tractable
way.
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