Mix-GENEO: A flexible filtration for multiparameter persistent homology detects digital images

Two important tasks in the field of topological data analysis (TDA) are building practical filtrations on objects and using TDA to detect the geometry and primarily topological structures. Motivated by these tasks, we have defined the difference between the two group equivariant non-expansive operators (GENEOs) by DGENEO and built multiparameter filtrations by operators on images named the multi-GENEO, multi-DGENEO, and mix-GENEO, and we proved the stability of both the interleaving distance and multiparameter persistence landscape of the multi-GENEO with respect to the pseudometric on images, modeled as bounded functions. We also gave an upper bound for the multi-DGENEO and mix-GENEO. In practical applications, we regarded the space of images on a discrete domain, and then we built multifiltrations on the discrete function space. Finally, we conducted a comparable experiment on the MNIST dataset to demonstrate that our bifiltrations are superior to 1-parameter filtrations. The experiment results demonstrate that our bifiltrations have the ability to detect geometric and topological differences of digital images.