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Monostatic SAR with Fold/Cusp Singularities

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Abstract

This paper analyses the image that one obtains by backprojecting synthetic aperture RADAR data collected on a flight-track with inflection points. The result is that one obtains artefacts that are of equal strength as the bona-fide part of the image. Furthermore, we obtain a weak normal form for operators associated to a fold/cusp canonical relation, which appears for our forward operator. Therefore this paper should be of use to researchers in different fields where such a structure arises.

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References

  1. Ambartsoumian, G., Felea, R., Krishnan, V., Nolan, C., Quinto, T.: A new class of singular Fourier intgral operators in synthetic aperture radar imaging. J. Funct. Anal. 264(1), 246–269 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Felea, R.: Composition calculus of Fourier integral operators with fold and blowdown singularities. Commu. Partial Differ. Equ. 30, 17171740 (2005)

    MathSciNet  Google Scholar 

  3. Felea, R., Greenleaf, A.: FIOs with open umbrellas and seismic inversion for cusp caustics. Math. Res. Lett. 17(5), 867–886 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities. Springer-Verlag, New York (1973)

    Book  MATH  Google Scholar 

  5. Greenleaf, A., Uhlmann, G.: Estimates for singular Radon transforms and pseudodifferential operators with singular symbols. J. Func. Anal. 89, 220232 (1990)

    Article  MathSciNet  Google Scholar 

  6. Greenleaf, A., Uhlman, G.: Compositions of some singular Fourier integral operators and estimates for restricted X-ray transforms. Ann. Inst. Fourier (Grenoble) 40(2), 443466 (1990)

    Article  Google Scholar 

  7. Greenleaf, A., Uhlman, G.: Compositions of some singular Fourier integral operators and estimates for restricted X-ray transforms II. Duke Math. J. 64, 415444 (1991)

    Article  Google Scholar 

  8. Hormander, L.: The Analysis of Linear Partial Differential Operators, vol. IV. Springer Verlag, New York (1983)

    Google Scholar 

  9. Krishnan, V.P., Quinto, E.T.: Microlocal aspects of bistatic synthetic aperture radar imaging. Inverse Probl. Imaging 5, 659–674 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Melrose, R., Uhlmann, G.: Lagrangian intersection and the Cauchy problem. Commun. Pure Appl. Math. 32(4), 483519 (1979)

    Article  MathSciNet  Google Scholar 

  11. Melrose, R., Taylor, M.: Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle. Adv. Math. 55(3), 242315 (1985)

    Article  MathSciNet  Google Scholar 

  12. Milnor, J.: Morse theory, Annals of Mathematics Studies, vol. 51. Princeton University Press, Princeton (1963)

    Google Scholar 

  13. Nolan, C.: Scattering in the presence of fold caustics. SIAM J. Appl. Math. 61, 659672 (2000)

    Article  MathSciNet  Google Scholar 

  14. Nolan, C., Cheney, M.: Microlocal analysis of synthetic aperture radar imaging. J. Fourier Anal. Appl. 10(2), 133–148 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

Both authors were supported by Raluca Felea’s Simons Foundation grant 209850.

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Authors and Affiliations

  1. School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY, 14623, USA

    Raluca Felea

  2. Department of Mathematics and Statistics, University of Limerick, Castletroy, Co., Limerick, Ireland

    Cliff Nolan

Authors
  1. Raluca Felea
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  2. Cliff Nolan
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Correspondence to Raluca Felea.

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Communicated by Eric Todd Quinto.

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Felea, R., Nolan, C. Monostatic SAR with Fold/Cusp Singularities. J Fourier Anal Appl 21, 799–821 (2015). https://doi.org/10.1007/s00041-015-9387-0

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  • DOI: https://doi.org/10.1007/s00041-015-9387-0

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