Automated Characterization and Monitoring of Material Shape using Riemannian Geometry

TL;DR

This paper introduces a Riemannian geometric framework for analyzing material shapes, enabling scalable, generalizable, and statistically rigorous shape characterization applicable to granular materials.

Contribution

It presents a novel application of Riemannian geometry to shape analysis, providing a scalable and generalizable method integrated with standard data analysis techniques.

Findings

Framework effectively characterizes granular material shapes

Enables statistical analysis like mean and variance of shapes

Demonstrates practical application on real dataset

Abstract

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.