Topological signal processing (TSP) over simplicial complexes typically assumes observations associated with the simplicial complexes are real scalars. In this paper, we develop TSP theories for the case where observations belong to general abelian groups, including function spaces that are commonly used to represent time-varying signals. Our approach generalizes the Hodge decomposition and allows for signal processing tasks to be performed on these more complex observations. We propose a unified and flexible framework for TSP that expands its applicability to a wider range of signal processing applications. Numerical results demonstrate the effectiveness of this approach and provide a foundation for future research in this area.

翻译：针对单纯复形的拓扑信号处理（TSP）通常假设与单纯复形相关的观测值为实数标量。本文针对观测值属于一般阿贝尔群（包括常用于表示时变信号的函数空间）的情形，发展了TSP理论。我们的方法推广了霍奇分解，使得能够在这些更复杂的观测值上执行信号处理任务。我们提出了一种统一且灵活的TSP框架，将其适用范围扩展到更广泛的信号处理应用。数值结果验证了该方法的有效性，并为该领域的未来研究奠定了基础。