The Impact of Changes in Resolution on the Persistent Homology of Images
This work addresses resolution selection challenges in topological data analysis for materials science, offering incremental improvements for specific applications.
The paper tackles the problem of selecting appropriate image resolution for persistent homology analysis by providing methods to choose the coarsest resolution that maintains results within a tolerance, based on prior information about functions, geometry, or density distributions, with numerical case studies on synthetic and porous material samples.
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.