Shape and Positional Geometry of Multi-Object Configurations
This work addresses the challenge of analyzing spatial relationships between objects in images for applications in computer vision and medical imaging, representing an incremental advancement in geometric modeling.
The paper tackles the problem of modeling multi-object configurations in 2D and 3D images by extending a medial/skeletal linking structure to capture positional properties, such as closeness and relative significance of objects, resulting in numerical invariants, a hierarchical ordering, and a proximity matrix for quantitative analysis.
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.