Vol. 5, No. 2, 2012

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The geodesic X-ray transform with fold caustics

Plamen Stefanov and Gunther Uhlmann

Vol. 5 (2012), No. 2, 219–260

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.

caustics, conjugate points, geodesic X-ray transform, integral geometry
Mathematical Subject Classification 2000
Primary: 53C65
Received: 20 April 2010
Revised: 16 February 2011
Accepted: 2 March 2011
Published: 27 August 2012
Plamen Stefanov
Department of Mathematics
Purdue University
150 N. University Street
West Lafayette, IN 47907-2607
United States
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