Modeling Multi-object Configurations via Medial/Skeletal Linking Structures

We introduce a method for modeling a configuration of objects in 2D or 3D images using a mathematical "skeletal linking structure" which will simultaneously capture the individual shape features of the objects and their positional information relative to one another. The objects may either have smooth boundaries and be disjoint from the others or share common portions of their boundaries with other objects in a piecewise smooth manner. These structures include a special class of "Blum medial linking structures", which are intrinsically associated to the configuration and build upon the Blum medial axes of the individual objects. We give a classification of the properties of Blum linking structures for generic configurations. The skeletal linking structures add increased flexibility for modeling configurations of objects by relaxing the Blum conditions and they extend in a minimal way the individual "skeletal structures" which have been previously used for modeling individual objects and capturing their geometric properties. This allows for the mathematical methods introduced for single objects to be significantly extended to the entire configuration of objects. These methods not only capture the internal shape structures of the individual objects but also the external structure of the neighboring regions of the objects. In the subsequent second paper (Damon and Gasparovic in Shape and positional geometry of multi-object configurations) we use these structures to identify specific external regions which capture positional information about neighboring objects, and we develop numerical measures for closeness of portions of objects and their significance for the configuration. This allows us to use the same mathematical structures to simultaneously analyze both the shape properties of the individual objects and positional properties of the configuration. This provides a framework for analyzing the statistical properties of collections of similar configurations such as for applications to medical imaging.