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Shape affects both the physical and chemical properties of a material.Characterizing the roughness, convexity, and general geometry of a material canyield 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 wheredifferent aspects of a material's geometry are emphasized over others, leadingto 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 showthat a Riemannian geometric framework for shape analysis is generalizable,computationally scalable, and can be directly integrated into common dataanalysis methods. In this framework, material shapes are abstracted as pointson a Riemannian manifold. This information can be used to construct statisticalmoments (e.g., means, variances) and perform tasks such as dimensionalityreduction and statistical process control. We provide a practical introductionto the mathematics of shape analysis through Riemannian geometry and illustrateits application on a manufactured/mined granular material dataset provided byCovia Corp. We show that the Riemannian framework can be used to automaticallyextract and quantify the shape of granular materials in a statisticallyrigorous manner.

Automated Characterization and Monitoring of Material Shape using Riemannian Geometry