Quantitative representations of relative positions and spatial relationships between image objects: A theoretical and algorithmic contribution
Research on the modeling of spatial relationships has been carried out for decades. We focus here on two issues: (a) How to quantitatively model a spatial relationship between two image objects; (b) How to represent a spatial relationship to a reference object. The force histogram, introduced earlier, provides a solution to the first question. It is a quantitative model of the position of an object relative to another, from which assessments of the spatial relationships between the objects can be deduced. In this work, we redefine the concept in a completely different but equivalent way. The new definition not only leads to a more efficient algorithm for computing force histograms with respect to raster data, but also is better adapted to the solving of important theoretical issues. We use it here to determine the behaviour of the force histogram toward any invertible affine transformation. To answer the second question, we introduce the notion of the force field-based map. It relies on the idea of considering a reference object as a physical entity that creates a force field around itself. Any directional relationship to the reference object can then be represented by a directional map, called force field-based map, derived from the force field. The representation, compared with existing ones, better copes with outliers, elongated objects and objects with complex shapes. Moreover, using the force field-based approach, directional spatial relationships between two objects can be modeled in such a way to satisfy some interesting properties. Finally, we show that there are strong theoretical connections between force fields and force histograms.