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.

生物对象的几何测量构成了许多定量分析的基础。Hausdorff 度量，例如物体的体积和面积，是对单个物体的简单而流行的描述；然而，对于大多数生物过程来说，物体之间的相互作用是不可忽视的，相邻物体的形状和功能是相互影响的。在本文中，我们提出了一种受K函数启发的对象之间几何相互作用的理论，用于空间点过程。我们的理论描述了两个对象之间的关系：参考对象和观察对象。我们生成参考对象的r 个平行集，计算r之间的交集- 平行集和观察对象，并在这些交叉点上定义度量。这些度量很简单，如体积或表面积，但描述了有关单个对象的形状及其成对几何关系的更多细节。最后，我们提出了一个汇总统计。为了评估这些措施，我们在公开可用的 FIB-SEM 3D 脑组织数据集中提出了细胞膜、线粒体、突触、囊泡和内质网的新分割，并使用我们提出的方法来分析此处的关键生物结构。