Automated characterization and monitoring of material shape using Riemannian geometry

Shape affects both the physical and chemical properties of a material. Characterizing the roughness, convexity, and general geometry of a material can yield information on its catalytic efficiency, solubility, elasticity, porosity, and overall effectiveness in the application of interest. However, material shape can be defined in a multitude of conflicting ways where different aspects of a material's geometry are emphasized over others, leading to bespoke measures of shape that are not easily generalizable. In this paper, we explore the use of Riemannian geometry in the analysis of shape, and show that a Riemannian geometric framework for shape analysis is generalizable, computationally scalable, and can be directly integrated into common data analysis methods. In this framework, material shapes are abstracted as points on a Riemannian manifold. This information can be used to construct statistical moments (e.g., means, variances) and perform tasks such as dimensionality reduction and statistical process control. We provide a practical introduction to the mathematics of shape analysis through Riemannian geometry and illustrate its application on a manufactured/mined granular material dataset provided by Covia Corp. We show that the Riemannian framework can be used to automatically extract and quantify the shape of granular materials in a statistically rigorous manner.