Mix-GENEO: A Flexible Filtration for Multiparameter Persistent Homology Detects Digital Images
This work addresses the challenge of improving image analysis in topological data analysis by introducing flexible multiparameter filtrations, though it appears incremental as it builds on existing filtration methods.
The paper tackled the problem of detecting geometric differences in digital images by developing multiparameter filtrations (multi-GENEO, multi-DGENEO, mix-GENEO) for topological data analysis, and demonstrated their superiority over 1-parameter filtrations on the MNIST dataset, such as distinguishing digits like 6 and 9.
Two important tasks in the field of Topological Data Analysis are building practical multifiltrations on objects and using TDA to detect the geometry. Motivated by the tasks, we build multiparameter filtrations by operators on images named multi-GENEO, multi-DGENEO and mix-GENEO, and we prove the stability of both the interleaving distance and multiparameter persistence landscape of multi-GENEO with respect to the pseudometric on bounded functions. We also give the estimations of upper bound for multi-DGENEO and mix-GENEO. In practical applications, we regard image as a discrete function space, and then we build multifiltrations on the discrete function space. Finally, we construct comparable experiment on MNIST dataset to demonstrate our bifiltrations are superior to 1-parameter filtrations including lower-star filtration and upper-star filtration. For instance, 6 and 9 can be distinguished by our bifiltrations, while they cannot be distinguished by 1-parameter filtrations. The experiment results demonstrate our bifiltrations have ability to detect geometric and topological differences of digital images.