Measuring Shape Relations Using r-Parallel Sets

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 inspired by K-functions for spatial point-processes. Our theory describes the relation between two objects: a reference and an observed object. We generate the r-parallel sets of the reference object, calculate the intersection between the r-parallel sets and the observed object, and define measures on these intersections. The measures are simple, like the volume or surface area, but describe further details about the shape of individual objects and their pairwise geometrical relation. Finally, we propose a summary-statistics. To evaluate these measures, we present a new segmentation of cell membrane, mitochondria, synapses, vesicles, and endoplasmic reticulum in a publicly available FIB-SEM 3D brain tissue data set and use our proposed method to analyze key biological structures herein.