A Multi-parameter Persistence Framework for Mathematical Morphology
This work provides a novel framework for image processing by combining mathematical morphology with topological data analysis, potentially automating structure optimization in images.
The paper tackles the problem of analyzing images by viewing morphological operations through persistent homology, demonstrating that they form a multiparameter filtration to extract topological and geometric information, and applies this framework to noisy binary, grayscale, and color images.
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.