Morse theory-based segmentation and fabric quantification of granular materials

Abstract

This article presents a robust Morse theory-1 based framework for segmenting 3D x-ray computed 2 tomography image (CT) and computing the fabric, rel-3 ative arrangement of particles, of granular ensembles. 4 The framework includes an algorithm for computing the 5 segmentation, a data structure for storing the segmenta-6 tion and representing both individual particles and the 7 connectivity network, and visualizations of topological 8 descriptors of the CT image that enable interactive ex-9 ploration. The Morse theory-based framework produces 10 superior quality segmentation of a granular ensemble as 11 compared to prior approaches based on the watershed 12 transform. The accuracy of the connectivity network 13 also improves. Further, the framework supports the effi-14 cient computation of various distribution statistics on 15 the segmentation and the connectivity network. Such 16 a comprehensive characterization and quantification of 17 the fabric of granular ensembles is the first step towards 18 a multiple length scale understanding of the behavior. 19 Keywords Granular ensemble • fabric • segmentation • 20 particle connectivity network • Morse theory • 21 Morse-Smale complex • topological persistence.

Towards Benchmark Data Generation for Feature Tracking in Scalar Fields

“…The function f is called Morse function if it satisfies the following two conditions: all critical points are non-degenerate, and they have distinct scalar values. One of the key observations is that any smooth function can be infinitesimally perturbed to satisfy these two conditions, and hence key results from Morse theory are widely applicable to robust analysis of scalar fields encountered in a variety of scientific domains [1,22,29]. The critical points of a Morse function can be classified based on the number of negative eigenvalues of the Hessian, also called the Morse index of the critical point.…”