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The geodesic X-ray
transform with fold caustics
Plamen Stefanov and Gunther Uhlmann |
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Vol. 5 (2012), No. 2, 219–260
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Abstract |
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We give a detailed microlocal study of X-ray transforms over geodesic-like families of curves with conjugate points of fold type. We show that the normal operator is the sum of a pseudodifferential operator and a Fourier integral operator. We compute the principal symbol of both operators and the canonical relation associated to the Fourier integral operator. In two dimensions, for the geodesic transform, we show that there is always a cancellation of singularities to some order, and we give an example where that order is infinite; therefore the normal operator is not microlocally invertible in that case. In the case of three dimensions or higher if the canonical relation is a local canonical graph we show microlocal invertibility of the normal operator. Several examples are also studied. |
Keywords
caustics, conjugate points, geodesic X-ray transform,
integral geometry
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Mathematical Subject Classification 2000
Primary: 53C65
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Milestones
Received: 20 April 2010
Revised: 16 February 2011
Accepted: 2 March 2011
Published: 27 August 2012
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Authors |
| Plamen Stefanov | |
| Department of Mathematics Purdue University 150 N. University Street West Lafayette, IN 47907-2607 United States |
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| http://www.math.purdue.edu/~stefanov/ | |
| Gunther Uhlmann | |
| Department of Mathematics University of Washington Seattle, WA 98195-4350 United States |
Department of Mathematics University of California at Irvine Irvine, CA 92697 United States |
| http://www.math.washington.edu/~gunther/ | |
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