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arXiv · 1402.5559

The geodesic ray transform on Riemannian surfaces with conjugate points

François Monard·Plamen Stefanov·Gunther Uhlmann
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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 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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François Monard, Plamen Stefanov, Gunther Uhlmann. 2014-03-26. The geodesic ray transform on Riemannian surfaces with conjugate points. https://doi.org/10.1007/s00220-015-2328-6

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