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.