Abstract
We consider the problem of sparse signal recovery in dynamic sensing scenarios. Specifically, we study the recovery of a sparse time-varying signal from linear measurements of a single static sensor that are taken at two different points in time. This setup can be modelled as observing a single signal using two different sensors – a real one and a virtual one induced by signal motion, and we examine the recovery properties of the resulting combined sensor. We show that not only can the signal be uniquely recovered with overwhelming probability by linear programming, but also the correspondence of signal values (signal motion) can be established between the two points in time. In particular, we show that in our scenario the performance of an undersampling static sensor is doubled or, equivalently, that the number of sufficient measurements of a static sensor is halved.
Acknowledgments. We gratefully acknowledge support by the German Science Foundation, grant GRK 1653.
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References
Abraham, I., Abraham, R., Bergounioux, M., Carlier, G.: Tomographic reconstruction from a few views: a multi-marginal optimal transport approach. Appl. Math. Optim. 75(1), 55–73 (2017)
Ferradans, S., Papadakis, N., Peyré, G., Aujol, J.: Regularized discrete optimal transport. SIAM J. Imaging Sci. 7(3), 1853–1882 (2014)
Foucart, S., Rauhut, H.: A Mathematical Introduction to Compressive Sensing. Springer, Heidelberg (2013)
Herman, G.T., Kuba, A.: Discrete Tomography: Foundations Algorithms and Applications. Birkhäuser, Basel (1999)
Lynch, K.P., Scarano, F.: An efficient and accurate approach to MTE-MART for time-resolved tomographic PIV. Exp. Fluids 56(3), 1–16 (2015)
Novara, M., Batenburg, K.J., Scarano, F.: Motion tracking-enhanced MART for tomographic PIV. Meas. Sci. Technol. 21(3), 035401 (2010)
Petra, S., Schnörr, C.: Average case recovery analysis of tomographic compressive sensing. Linear Algebra Appl. 441, 168–198 (2014)
Puy, G., Vandergheynst, P.: Robust image reconstruction from multiview measurements. SIAM J. Imaging Sci. 7(1), 128–156 (2014)
Saumier, L., Khouider, B., Agueh, M.: Optimal transport for particle image velocimetry. Commun. Math. Sci. 13(1), 269–296 (2015)
Saumier, L., Khouider, B., Agueh, M.: Optimal transport for particle image velocimetry: real data and postprocessing algorithms. SIAM J. Appl. Math. 75(6), 2495–2514 (2015)
Schanz, D., Gesemann, S., Schröder, A.: Shake-the-box: lagrangian particle tracking at high particle image densities. Exp. Fluids 57(70), 1–27 (2016)
Villani, C.: Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften. Springer, Heidelberg (2009)
Xu, Y., Yin, W.: A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion. SIAM J. Imaging Sci. 6(3), 1758–1789 (2013)
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Dalitz, R., Petra, S., Schnörr, C. (2017). Compressed Motion Sensing. In: Lauze, F., Dong, Y., Dahl, A. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2017. Lecture Notes in Computer Science(), vol 10302. Springer, Cham. https://doi.org/10.1007/978-3-319-58771-4_48
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DOI: https://doi.org/10.1007/978-3-319-58771-4_48
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