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Microlocal Analysis of Elliptical Radon Transforms with Foci on a Line

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Book cover The Mathematical Legacy of Leon Ehrenpreis

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 16))

Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. Tata Institute of Fundamental Research Centre for Applicable Mathematics, Bangalore, 560065, India

    Venkateswaran P. Krishnan

  2. 1759 Beechwood Blvd., Pittsburgh, PA, 15217, USA

    Howard Levinson

  3. Tufts University, Medford, MA, 02155, USA

    Eric Todd Quinto

Authors
  1. Venkateswaran P. Krishnan
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  2. Howard Levinson
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  3. Eric Todd Quinto
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Corresponding author

Correspondence to Eric Todd Quinto .

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Editors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, Milano, 20133, Italy

    Irene Sabadini

  2. Schmid College of Science and Technology, Chapman University, University Dr. 1, Orange, 92868, California, USA

    Daniele C Struppa

Additional information

We dedicate this article to the memory of Leon Ehrenpreis, a brilliant mathematician and a Mensch.

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Krishnan, V.P., Levinson, H., Quinto, E.T. (2012). Microlocal Analysis of Elliptical Radon Transforms with Foci on a Line. In: Sabadini, I., Struppa, D. (eds) The Mathematical Legacy of Leon Ehrenpreis. Springer Proceedings in Mathematics, vol 16. Springer, Milano. https://doi.org/10.1007/978-88-470-1947-8_11

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