Topological Estimation of 2D Image Data via Subsampling and Application to Firn

Victoria Belotti,Matthew Jester,Sarah Day,Yu-Min Chung,Kaitlin Keegan

doi:10.1109/access.2021.3050377

Abstract

We develop a novel statistical approach to estimate topological information from large, noisy images. Our main motivation is to measure pore microstructure in 2-dimensional X-ray micro-computed tomography (micro-CT) images of ice cores at different depths. The pore space in these samples is where gas can move and get trapped within the ice column and is of interest to climate scientists. While the field of topological data analysis offers tools (e.g. lifespan cutoff and PD Thresholding) for estimating topological information in noisy images, direct application of these techniques to large images often leads to inaccuracies and proves infeasible as image size and noise levels grow. Our approach uses image subsampling to estimate the number of holes of a prescribed size range in a computationally feasible manner. In applications where holes naturally have a known size range on a smaller scale than the full image, this approach offers a means of estimating Betti numbers, or global counts of holes of various dimensions, via subsampling of the image.

Highlights

Topological data analysis (TDA) is an emerging field in applied mathematics that offers techniques for studying topological features such as connected components and holes in point cloud and image data
We focus on two key ways of improving estimates of counts of features when using Persistence Diagram (PD) Thresholding to study large, noisy firn data, and, by extension, similar imaging problems in materials science
The 0-th Betti number, β0, counts the number of 0-dimensional holes or connected components and β1 counts the number of 1-dimensional holes in the 2-dimensional image corresponding to tunnels

Summary

Introduction

Topological data analysis (TDA) is an emerging field in applied mathematics that offers techniques for studying topological features such as connected components and holes in point cloud and image data (see e.g. review and survey articles [1]–[4]). It has proven successful in many scientific applications (see [5] that includes a list of applications and references therein [1]–[4]). The firn data we have consists of 3-dimensional grayscale image data, of which we can use 2-dimensional cross-sectional image slices (see FIGURE 1 for sample image slices) in order to test our methods

Objectives

Methods

Conclusion