MEDIAL/SKELETAL LINKING STRUCTURES FOR MULTI-REGION CONFIGURATIONS

Abstract. We consider a generic configuration of regions, consisting of a collection of distinct compact regions {Ωi} in Rn+1 which may be either smooth regions disjoint from the others or regions which meet on their piece-wise smooth boundaries Bi in a generic way. We introduce a skeletal linking structure for the collection of regions which simultaneously captures the re-gions ’ individual shapes and geometric properties as well as the “positional geometry”of the collection. The linking structure extends in a minimal way the individual “skeletal structures”on each of the regions. This allows us to significantly extend the mathematical methods introduced for single regions to the configuration of regions. We prove for a generic configuration of regions the existence of a special type of Blum linking structure which builds upon the Blum medial axes of the individual regions. This requires proving several transversality theorems for certain associated “multi-distance ” and “height-distance ” functions for such configurations. We show that by relaxing the conditions on the Blum linking structures we obtain the more general class of skeletal linking structures which still capture the geometric properties. The skeletal linking structure is used to analyze the “positional geometry”of the configuration. This involves using the “linking flow ” to identify neighbor-hoods of the configuration regions which capture their positional relations. As well as yielding geometric invariants which capture the shapes and geometry of individual regions, these structures are used to define invariants which measure positional properties of the configuration such as: measures of relative close-ness of neighboring regions and relative significance of the individual regions for the configuration. All of these invariants are computed by formulas involving “skeletal linking integrals”on the internal skeletal structures of the regions. These invariants are then used to construct a “tiered linking graph”, which for given thresholds of closeness and/or significance, identifies subconfigurations and provides a hierarchical ordering in terms of order of significance. 1