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The identification problem for the attenuated X-ray transform
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 136, Number 5, October 2014
- pp. 1215-1247
- 10.1353/ajm.2014.0035
- Article
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We study the problem of recovery both the attenuation $a$ and the source $f$ in the attenuated X-ray
transform in the plane. We study the linearization as well. It turns out that there is a natural
Hamiltonian flow that determines which singularities we can recover. If the perturbation $\delta a$
is supported in a compact set that is non-trapping for that flow, then the problem is well posed.
Otherwise, it may not be, and least in the case of radial $a$, $f$, it is not. We present uniqueness
and non-uniqueness results for both the linearized and the non-linear problem; as well as a H\"older
stability estimate.