The geodesic X-ray transform with fold caustics

By Plamen Stefanov, Gunther Uhlmann. 2010

63 citations recorded by the source corpus.

We give a detailed microlocal study of X-ray transforms over geodesics-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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/deea5eab930745a537e63ea308ddadb8abc62ca5

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

The Calderón problem in transversally anisotropic geometries (2013, background): “See also [2], [40] for microlocal analysis of the ray transform in some non-simple geometries.”
The inverse problem for the local geodesic ray transform (2012, background): “In [19] one can find a microlocal study of the geodesic X-ray transform with fold caustics.”
Sensitivity analysis of an inverse problem for the wave equation with caustics (2012, methodology): “Indeed, in that case, it is shown in [35] that the canonical relation associated with the operator germ N2,ξ∗ is locally a canonical graph and hence N2,ξ∗ is bounded from L (2)(Ω 0 ,R ) to H d 2 (U(x∗),R ) for some neighborhood U(x∗) of x∗.”