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Compressed Motion Sensing

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Scale Space and Variational Methods in Computer Vision (SSVM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 10302))

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

  1. 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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ferradans, S., Papadakis, N., Peyré, G., Aujol, J.: Regularized discrete optimal transport. SIAM J. Imaging Sci. 7(3), 1853–1882 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Foucart, S., Rauhut, H.: A Mathematical Introduction to Compressive Sensing. Springer, Heidelberg (2013)

    Book  MATH  Google Scholar 

  4. Herman, G.T., Kuba, A.: Discrete Tomography: Foundations Algorithms and Applications. Birkhäuser, Basel (1999)

    Book  MATH  Google Scholar 

  5. 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)

    Google Scholar 

  6. Novara, M., Batenburg, K.J., Scarano, F.: Motion tracking-enhanced MART for tomographic PIV. Meas. Sci. Technol. 21(3), 035401 (2010)

    Article  Google Scholar 

  7. Petra, S., Schnörr, C.: Average case recovery analysis of tomographic compressive sensing. Linear Algebra Appl. 441, 168–198 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Puy, G., Vandergheynst, P.: Robust image reconstruction from multiview measurements. SIAM J. Imaging Sci. 7(1), 128–156 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Saumier, L., Khouider, B., Agueh, M.: Optimal transport for particle image velocimetry. Commun. Math. Sci. 13(1), 269–296 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Schanz, D., Gesemann, S., Schröder, A.: Shake-the-box: lagrangian particle tracking at high particle image densities. Exp. Fluids 57(70), 1–27 (2016)

    Google Scholar 

  12. Villani, C.: Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften. Springer, Heidelberg (2009)

    Book  Google Scholar 

  13. 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)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Stefania Petra .

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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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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-58770-7

  • Online ISBN: 978-3-319-58771-4

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