  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.
