Title: Shape and Positional Geometry of Multi-Object Configurations,
Abstract: In previous work, we introduced a method for modeling a configuration of
objects in 2D and 3D images using a mathematical "medial/skeletal linking
structure." In this paper, we show how these structures allow us to capture
positional properties of a multi-object configuration in addition to the shape
properties of the individual objects. In particular, we introduce numerical
invariants for positional properties which measure the closeness of neighboring
objects, including identifying the parts of the objects which are close, and
the "relative significance" of objects compared with the other objects in the
configuration. Using these numerical measures, we introduce a hierarchical
ordering and relations between the individual objects, and quantitative
criteria for identifying subconfigurations. In addition, the invariants provide
a "proximity matrix" which yields a unique set of weightings measuring overall
proximity of objects in the configuration. Furthermore, we show that these
invariants, which are volumetrically defined and involve external regions, may
be computed via integral formulas in terms of "skeletal linking integrals"
defined on the internal skeletal structures of the objects.