Microlocal Analysis of Elliptical Radon Transforms with Foci on a Line

  • Venkateswaran P. Krishnan
  • Howard Levinson
  • Eric Todd Quinto
Part of the Springer Proceedings in Mathematics book series (PROM, volume 16)


In this paper, we take a microlocal approach to the study of an integral geometric problem involving integrals of a function on the plane over two-dimensional sets of ellipses on the plane. We focus on two cases: (a) the family of ellipses where one focus is fixed at the origin and the other moves along the x-axis, and (b) the family of ellipses having a common offset geometry.

For case (a), we characterize the Radon transform as a Fourier integral operator associated to a fold and blowdown. This has implications on how the operator adds singularities, how backprojection reconstructions will show those singularities, and in comparison of the strengths of the original and added singularities in a Sobolev sense.

For case (b), we show that this Radon transform has similar structure to case (a): it is a Fourier integral operator associated to a fold and blowdown. This case is related to previous results of authors one and three. We characterize singularities that are added by the reconstruction operator, and we present reconstructions from the authors’ algorithm that illustrate the microlocal properties.


Synthetic Aperture Radar Synthetic Aperture Radar Imaging Flight Path Fourier Integral Operator Reconstruction Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



All three authors were supported by Quinto’s NSF grant DMS 0908015. Krishnan was supported by summer postdoctoral supplements to Quinto’s NSF grant DMS 0908015 in 2010 (DMS 1028096) and 2011 (DMS 1129154), and by NSF grant DMS 1109417. Levinson received REU support.


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Copyright information

© Springer-Verlag Italia 2012

Authors and Affiliations

  • Venkateswaran P. Krishnan
    • 1
  • Howard Levinson
    • 2
  • Eric Todd Quinto
    • 3
  1. 1.Tata Institute of Fundamental Research Centre for Applicable MathematicsBangaloreIndia
  2. 2.PittsburghUSA
  3. 3.Tufts UniversityMedfordUSA

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