MMPersistence: A mathematical morphology-oriented software library for computing persistent homology on cubical complexes

TL;DR

MMPersistence is a software library that combines mathematical morphology operations with persistent homology computations on cubical complexes to analyze digital images' topological and morphological features at multiple scales.

Contribution

It introduces a novel framework integrating morphological operations with persistent homology, enriching topological analysis with local geometric information in digital images.

Findings

Enables multiscale topological analysis of images.

Provides richer local geometric information than traditional cubical homology.

Unifies morphological processing with topological data analysis.

Abstract

Mathematical morphology (MM) is a powerful and widely used framework in image processing. Through set-theoretic and discrete geometric principles, MM operations such as erosion, dilation, opening, and closing effectively manipulate digital images by modifying local structures via structuring elements (SEs), while cubical homology captures global topological features such as connected components and loop structures within images. Building on the GUDHI package for persistent homology (PH) computation on cubical complexes, we propose the MMPersistence library, which integrates MM operations with diverse SEs and PH computation to extract multiscale persistence information. By employing SEs of different shapes to construct topological filtrations, the proposed MM-based PH framework encodes both spatial and morphological characteristics of digital images, providing richer local geometric information than conventional cubical homology alone and establishing a unified foundation for analyzing digital images that integrates topological insight with morphological image processing techniques.