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In previous work, we introduced a method for modeling a configuration ofobjects in 2D and 3D images using a mathematical "medial/skeletal linkingstructure." In this paper, we show how these structures allow us to capturepositional properties of a multi-object configuration in addition to the shapeproperties of the individual objects. In particular, we introduce numericalinvariants for positional properties which measure the closeness of neighboringobjects, including identifying the parts of the objects which are close, andthe "relative significance" of objects compared with the other objects in theconfiguration. Using these numerical measures, we introduce a hierarchicalordering and relations between the individual objects, and quantitativecriteria for identifying subconfigurations. In addition, the invariants providea "proximity matrix" which yields a unique set of weightings measuring overallproximity of objects in the configuration. Furthermore, we show that theseinvariants, which are volumetrically defined and involve external regions, maybe computed via integral formulas in terms of "skeletal linking integrals"defined on the internal skeletal structures of the objects.

Shape and Positional Geometry of Multi-Object Configurations