Measuring shape relations using r-parallel sets

TL;DR

This paper introduces a new geometrical framework using r-parallel sets to quantify shape relations and interactions between biological objects, enhancing traditional measures like volume and area.

Contribution

It develops a novel theory for measuring shape interactions based on spatial point processes and proposes summary statistics for collections of shapes.

Findings

Effective measures for shape interaction were demonstrated on 3D biological data.

The proposed measures capture detailed shape and interaction information beyond traditional descriptors.

The framework provides new insights into geometrical relationships in biological structures.

Abstract

Geometrical measurements of biological objects form the basis of many quantitative analyses. Hausdorff measures such as the volume and the area of objects are simple and popular descriptors of individual objects, however, for most biological processes, the interaction between objects cannot be ignored, and the shape and function of neighboring objects are mutually influential. In this paper, we present a theory on the geometrical interaction between objects based on the theory of spatial point processes. Our theory is based on the relation between two objects: a reference and an observed object. We generate the $r$-parallel sets of the reference object, we calculate the intersection between the $r$-parallel sets and the observed object, and we define measures on these intersections. Our measures are simple like the volume and area of an object, but describe further details about the shape of individual objects and their pairwise geometrical relation. Finally, we propose a summary statistics for collections of shapes and their interaction. We evaluate these measures on a publicly available FIB-SEM 3D data set of an adult rodent.