基于Minkowski张量的随机网格胞形分析
Cell shape analysis of random tessellations based on Minkowski tensors
摘要：
在何种程度上是一个镶嵌特征的随机过程，产生他们的单个细胞的形状指数？在随机几何学和无序材料物理学的背景下，这对应于不同随机模型之间的关系问题。在合成材料和生物材料的图像分析的背景下，这个问题是中央的问题推断信息的形成过程中产生的随机结构的空间测量。我们解决这个问题的理论为基础的模拟研究来自闵可夫斯基张量的各种镶嵌模型的形状指数。我们专注于两个指数之间的关系：一个等周比的经验平均值的细胞体积和面积和细胞的伸长率量化的特征值比的界面Minkowski张量。这些量的模拟数据，以及其分布和细胞形状和体积的相关性，提出了Voronoi马赛克的泊松点过程，行列式和永久点过程，吉布斯硬核和随机顺序吸收过程，以及拉盖尔镶嵌的多分散球体和STIT-和泊松超平面镶嵌。这些数据是补充机械稳定的结晶球和无序的椭球填料和面积最小化泡沫模型。我们发现，单个细胞的形状指数是不足以明确识别的生成过程中，即使在这有限的一组过程。然而，我们发现许多这些镶嵌模型之间的形状指数的显着差异。给定镶嵌的实现，这些形状指数可以缩小可能的生成过程的选择，提供了一个强大的工具，可以通过密度分辨体积形状相关性进一步加强。
英文摘要：
To which degree are shape indices of individual cells of a tessellation characteristic for the stochastic process that generates them? Within the context of stochastic geometry and the physics of disordered materials, this corresponds to the question of relationships between different stochastic models. In the context of image analysis of synthetic and biological materials, this question is central to the problem of inferring information about formation processes from spatial measurements of resulting random structures. We address this question by a theory-based simulation study of shape indices derived from Minkowski tensors for a variety of tessellation models. We focus on the relationship between two indices: an isoperimetric ratio of the empirical averages of cell volume and area and the cell elongation quantified by eigenvalue ratios of interfacial Minkowski tensors. Simulation data for these quantities, as well as for distributions thereof and for correlations of cell shape and volume, are presented for Voronoi mosaics of the Poisson point process, determinantal and permanental point processes, and Gibbs hard-core and random sequential absorption processes as well as for Laguerre tessellations of polydisperse spheres and STIT- and Poisson hyperplane tessellations. These data are complemented by mechanically stable crystalline sphere and disordered ellipsoid packings and area-minimising foam models. We find that shape indices of individual cells are not sufficient to unambiguously identify the generating process even amongst this limited set of processes. However, we identify significant differences of the shape indices between many of these tessellation models. Given a realization of a tessellation, these shape indices can narrow the choice of possible generating processes, providing a powerful tool which can be further strengthened by density-resolved volume-shape correlations.
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