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The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points

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Abstract

We study the geodesic X-ray transform X on compact Riemannian surfaces with conjugate points. Regardless of the type of the conjugate points, we show that we cannot recover the singularities and, therefore, this transform is always unstable (ill-posed). We describe the microlocal kernel of X and relate it to the conjugate locus. We present numerical examples illustrating the cancellation of singularities. We also show that the attenuated X-ray transform is well posed if the attenuation is positive and there are no more than two conjugate points along each geodesic; but it is still ill-posed if there are three or more conjugate points. Those results follow from our analysis of the weighted X-ray transform.

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Authors and Affiliations

  1. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    François Monard & Gunther Uhlmann

  2. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Plamen Stefanov

Authors
  1. François Monard
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  2. Plamen Stefanov
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  3. Gunther Uhlmann
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Correspondence to Gunther Uhlmann.

Additional information

Communicated by S. Zelditch

First author partly supported by NSF Grant No. 1265958.

Second author partly supported by a NSF Grant DMS-1301646.

Third author partly supported by NSF Grant No. 1265958 and a Simons Fellowship.

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Monard, F., Stefanov, P. & Uhlmann, G. The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points. Commun. Math. Phys. 337, 1491–1513 (2015). https://doi.org/10.1007/s00220-015-2328-6

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  • DOI: https://doi.org/10.1007/s00220-015-2328-6

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