# Shape and Positional Geometry of Multi-Object Configurations

## TL;DR

This paper extends skeletal linking structures to quantify positional relationships among multiple objects in 2D and 3D images, enabling hierarchical and proximity analyses based on volumetric invariants.

## Contribution

It introduces numerical invariants for positional properties, hierarchical orderings, and a proximity matrix derived from skeletal linking integrals, advancing multi-object configuration analysis.

## Key findings

- Numerical invariants effectively measure object proximity and significance.
- Hierarchical and relational structures enable detailed configuration analysis.
- Proximity matrices provide comprehensive measures of object closeness.

## 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.