HV geometry for signal comparison

Han, Ruiyu; Slepčev, Dejan; Yang, Yunan

doi:10.1090/qam/1672

Authors: Ruiyu Han, Dejan Slepčev and Yunan Yang
Journal: Quart. Appl. Math. 82 (2024), 391-430
MSC (2020): Primary 49J45, 58E10, 58E30, 65D18; Secondary 49K40, 58E50
DOI: https://doi.org/10.1090/qam/1672
Published electronically: June 16, 2023
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Abstract: In order to compare and interpolate signals, we investigate a Riemannian geometry on the space of signals. The metric allows discontinuous signals and measures both horizontal (thus providing many benefits of the Wasserstein metric) and vertical deformations. Moreover, it allows for signed signals, which overcomes the main deficiency of optimal transportation-based metrics in signal processing. We characterize the metric properties of the space of signals and establish the regularity and stability of geodesics. Furthermore, we introduce an efficient numerical scheme to compute the geodesics and present several experiments which highlight the nature of the metric.

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