The Impact of Changes in Resolution on the Persistent Homology of Images

TL;DR

This paper investigates how changing image resolution affects persistent homology analysis, providing methods to select the coarsest resolution that preserves structural information within acceptable limits.

Contribution

It introduces methods to determine the optimal resolution for persistent homology analysis based on prior information, applicable even when theoretical bounds are unknown.

Findings

Methods for resolution selection based on prior information

Numerical case studies on synthetic and porous material images

Guidelines for balancing resolution and data requirements

Abstract

Digital images enable quantitative analysis of material properties at micro and macro length scales, but choosing an appropriate resolution when acquiring the image is challenging. A high resolution means longer image acquisition and larger data requirements for a given sample, but if the resolution is too low, significant information may be lost. This paper studies the impact of changes in resolution on persistent homology, a tool from topological data analysis that provides a signature of structure in an image across all length scales. Given prior information about a function, the geometry of an object, or its density distribution at a given resolution, we provide methods to select the coarsest resolution yielding results within an acceptable tolerance. We present numerical case studies for an illustrative synthetic example and samples from porous materials where the theoretical bounds are unknown.