Causal Linear Topological Filters over a 2-Simplex

TL;DR

This paper extends topological filters to causal signal flow over a 2-simplex, introducing a concurrent extension to resolve conflicts and singularities, thereby generalizing classical filter theory using sheaves on simplicial complexes.

Contribution

It develops a novel framework for causal topological filters on 2-simplices, including methods to resolve conflicts and singularities in sheaf assignments.

Findings

Extended sheaf-based filters to 2-simplices

Resolved conflicts via concurrent extension with an auxiliary 1-simplex

Addressed singularities from double cone connections

Abstract

Topological filters via sheaves generalize the classical linear translation-invariant filter theory by attaching the filter computation locally to a simplicial topological space. This paper develops topological filters for causal signal flow over a 2-simplex. Our construction retains the established construction for 1-simplices and we show how an apparent conflict in the sheaf assignment can be resolved by a concurrent extension, which introduces an auxiliary 1-simplex that computes the resolution. Furthermore, we discuss how singularities formed by double cone connections can be resolved.