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Remote Sensing via ℓ 1-Minimization

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Abstract

We consider the problem of detecting the locations of targets in the far field by sending probing signals from an antenna array and recording the reflected echoes. Drawing on key concepts from the area of compressive sensing, we use an ℓ 1-based regularization approach to solve this, generally ill-posed, inverse scattering problem. As is common in compressive sensing, we exploit randomness, which in this context comes from choosing the antenna locations at random. With n antennas we obtain n 2 measurements of a vector \(x \in\mathbb{C}^{N}\) representing the target locations and reflectivities on a discretized grid. It is common to assume that the scene x is sparse due to a limited number of targets. Under a natural condition on the mesh size of the grid, we show that an s-sparse scene can be recovered via ℓ 1-minimization with high probability if n 2≥Cslog2(N). The reconstruction is stable under noise and when passing from sparse to approximately sparse vectors. Our theoretical findings are confirmed by numerical simulations.

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Acknowledgements

M.H. and H.R. acknowledge support by the Hausdorff Center for Mathematics and by the ERC Starting Grant SPALORA StG 258926. T.S. was supported by the National Science Foundation and DTRA under grant DTRA-DMS 1042939, and by DARPA under grant N66001-11-1-4090. Parts of this manuscript have been written during a stay of H.R. at the Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis. T.S. thanks Haichao Wang for useful comments on an early version of this manuscript. The authors also wish to thank Axel Obermeier and the anonymous reviewers for helpful comments and corrections.

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Authors and Affiliations

  1. Hausdorff Center for Mathematics and Institute for Numerical Simulation, University of Bonn, Bonn, Germany

    Max Hügel

  2. Lehrstuhl C für Mathematik (Analysis), RWTH Aachen University, Aachen, Germany

    Holger Rauhut

  3. Department of Mathematics, University of California at Davis, Davis, CA, USA

    Thomas Strohmer

Authors
  1. Max Hügel
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  2. Holger Rauhut
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  3. Thomas Strohmer
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Correspondence to Holger Rauhut.

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Hügel, M., Rauhut, H. & Strohmer, T. Remote Sensing via ℓ 1-Minimization. Found Comput Math 14, 115–150 (2014). https://doi.org/10.1007/s10208-013-9157-9

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  • DOI: https://doi.org/10.1007/s10208-013-9157-9

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