Topology-based Filtering of Graph Signals via Persistent Homology
We study topology-based filtering of vertex-defined signals on graphs and their two-dimensional analogues. Unlike graph-spectral filters, the proposed approach distinguishes features by topological persistence rather than by spatial wavelength or periodicity. We consider graphs with faces embedded in surfaces, a class that includes discrete models of images and meshes. We prove that, in general, exact simultaneous removal of low-persistence features in dimensions $0$ and $1$ is impossible. This motivates a relaxed formulation, for which we introduce the Low Persistence Filter (LPF). The LPF removes finite-persistence features below a prescribed threshold while controlling the resulting $\ell_\infty$ perturbation of the signal. We illustrate the method on one-dimensional signals, two-dimensional images, and signals on triangular meshes. A Python implementation is publicly available.