边空间中的流平滑与去噪：图信号处理
Flow Smoothing and Denoising: Graph Signal Processing in the Edge-Space
AI总结：
本文研究了图信号处理中针对图边数据的平滑与去噪问题，提出基于边拉普拉斯算子的新型滤波器以解决传统图拉普拉斯算子在处理流信号时的不足，并通过伦敦街道网络上的合成交通流去噪实验验证了方法的有效性。
摘要：
本文聚焦于设计用于处理定义在图边上的数据的图信号处理工具。我们首先表明，传统的图信号处理工具可能不适用于此类信号的分析。具体而言，我们讨论了从（通常考虑的变种）图拉普拉斯算子继承的`平滑信号`的概念在处理编码了流概念的边信号时并不适用。为克服这一限制，我们引入了一类基于边拉普拉斯算子（图简复形阶数为一的霍奇拉普拉斯算子的特例）的滤波器。我们展示了该边拉普拉斯算子如何产生低通滤波器，以在处理后的信号中强制（近似）流量守恒。此外，我们还展示了这些新滤波器如何与更经典的基于拉普拉斯的线图处理方法相结合。最后，我们通过去噪伦敦街道网络上的合成交通流来示例所开发的工具。
英文摘要：
This paper focuses on devising graph signal processing tools for the treatment of data defined on the edges of a graph. We first show that conventional tools from graph signal processing may not be suitable for the analysis of such signals. More specifically, we discuss how the underlying notion of a `smooth signal' inherited from (the typically considered variants of) the graph Laplacian are not suitable when dealing with edge signals that encode a notion of flow. To overcome this limitation we introduce a class of filters based on the Edge-Laplacian, a special case of the Hodge-Laplacian for simplicial complexes of order one. We demonstrate how this Edge-Laplacian leads to low-pass filters that enforce (approximate) flow-conservation in the processed signals. Moreover, we show how these new filters can be combined with more classical Laplacian-based processing methods on the line-graph. Finally, we illustrate the developed tools by denoising synthetic traffic flows on the London street network.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
