A Multi-parameter Persistence Framework for Mathematical Morphology

TL;DR

This paper introduces a novel framework that applies multi-parameter persistent homology to mathematical morphology, enabling enhanced analysis of image topology and geometry, and automating image structure optimization.

Contribution

It presents a new multiparameter persistence framework for mathematical morphology, bridging topological data analysis with image processing techniques.

Findings

Effective analysis of noisy binary, grayscale, and color images.

Automated methods for optimizing image structure analysis.

Demonstrated the framework's ability to extract topological and geometric information.

Abstract

The field of mathematical morphology offers well-studied techniques for image processing. In this work, we view morphological operations through the lens of persistent homology, a tool at the heart of the field of topological data analysis. We demonstrate that morphological operations naturally form a multiparameter filtration and that persistent homology can then be used to extract information about both topology and geometry in the images as well as to automate methods for optimizing the study and rendering of structure in images. For illustration, we apply this framework to analyze noisy binary, grayscale, and color images.