An emerging way to deal with high-dimensional non-euclidean data is to assume that the underlying structure can be captured by a graph. Recently, ideas have begun to emerge related to the analysis of time-varying graph signals. This work aims to elevate the notion of joint harmonic analysis to a full-fledged framework denoted as Time-Vertex Signal Processing, that links together the time-domain signal processing techniques with the new tools of graph signal processing. This entails three main contributions: (a) We provide a formal motivation for harmonic time-vertex analysis as an analysis tool for the state evolution of simple Partial Differential Equations on graphs. (b) We improve the accuracy of joint filtering operators by up-to two orders of magnitude. (c) Using our joint filters, we construct time-vertex dictionaries analyzing the different scales and the local time-frequency content of a signal. The utility of our tools is illustrated in numerous applications and datasets, such as dynamic mesh denoising and classification, still-video inpainting, and source localization in seismic events. Our results suggest that joint analysis of time-vertex signals can bring benefits to regression and learning.

The exploitation of offshore wind resources is a crucial step towards a clean energy future. It requires an advanced approach for high-resolution wind resource evaluations. We explored the suitability of the Sentinel-1 Level-2 OCN ocean wind field (OWI) product for offshore wind resource assessments. The SAR data were compared to in situ observations and three reanalysis products: the global reanalysis ERA5 and two regional reanalyses CARRA and NORA3. This case study matches 238 scenes from 2022 for the Goliat station, an oil platform located 85 km northwest of Hammerfest in the Barents Sea, where a new offshore wind park has been proposed. The analysis showed that despite their unique limitations in spatial and temporal resolutions, all data sources have similar statistical properties (RMSE, correlation coefficient, and standard deviation). The Weibull parameters characterizing the wind speed distributions showed strong similarities between the Sentinel-1 and all reanalysis data. The Weibull parameters of the in situ measurements showed an underestimation of wind speed compared to all other sources. Comparing the full reanalysis datasets with the subsets matching the SAR scenes, only slight changes in Weibull parameters were found, indicating that, despite its low temporal resolution, the Sentinel-1 Level 2 OWI product can compete with the more commonly used reanalysis products in the estimation of offshore wind resources. Its high spatial resolution, which is unmatched by other methods, renders it especially valuable in offshore areas close to complex coastlines and in resolving weather events at a smaller scale.