On the microlocal analysis of the geodesic X-ray transform with conjugate points

Sean Holman; Gunther Uhlmann

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arXiv · 1502.06545

On the microlocal analysis of the geodesic X-ray transform with conjugate points

Sean Holman·Gunther Uhlmann

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Abstract

We study the microlocal properties of the geodesic X-ray transform $\mathcal{X}$ on a manifold with boundary allowing the presence of conjugate points. Assuming that there are no self-intersecting geodesics and all conjugate pairs are nonsingular we show that the normal operator $\mathcal{N} = \mathcal{X}^t \circ \mathcal{X}$ can be decomposed as the sum of a pseudodifferential operator of order $-1$ and a sum of Fourier integral operators. We also apply this decomposition to prove inversion of $\mathcal{X}$ is only mildly ill-posed in dimension three or higher.

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Sean Holman, Gunther Uhlmann. 2015-02-23. On the microlocal analysis of the geodesic X-ray transform with conjugate points. https://arxiv.org/abs/1502.06545

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