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1. Double b-fibrations and desingularization of the X-ray transform on manifolds with strictly convex boundary

   https://doi.org/10.5802/jep.266  

   Mazzeo, Rafe; Monard, François (January 2024, Journal de l’École polytechnique — Mathématiques)

   We study the mapping properties of the X-ray transform and its adjoint on spaces of conormal functions on Riemannian manifolds with strictly convex boundary. After desingularizing the double fibration, and expressing the X-ray transform and its adjoint using b-fibrations operations, we employ tools related to Melrose’s pushforward theorem to describe the mapping properties of these operators on various classes of polyhomogeneous functions, with special focus to computing how leading order coefficients are transformed. The appendix explains that a naive use of the pushforward theorem leads to a suboptimal result with non-sharp index sets. Our improved results are obtained by closely inspecting Mellin functions which arise in the process, showing that certain coefficients vanish. This recovers some sharp results known by other methods. A number of consequences for the mapping properties of the X-ray transform and its normal operator(s) follow. 

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2. Inverting the local geodesic ray transform of higher rank tensors

   https://doi.org/10.1088/1361-6420/ab1ace  

   de Hoop, Maarten V; Uhlmann, Gunther; Zhai, Jian (January 2018, Inverse Problems)

   Consider a Riemannian manifold in dimension n ≥ 3 with strictly convex boundary. We prove the local invertibility, up to potential fields, of the geodesic ray transform on tensor fields of rank four near a boundary point. This problem is closely related with elastic qP-wave tomography. Under the condition that the manifold can be foliated with a continuous family of strictly convex hypersurfaces, the local invertibility implies a global result. One can straightforwardly adapt the proof to show similar results for tensor fields of arbitrary rank. 

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3. Mixed ray transform on simple $$2$$-dimensional Riemannian manifolds

   https://doi.org/10.1090/proc/14601  

   de Hoop, Maarten V.; Saksala, Teemu; Zhai, Jian (January 2019, Proceedings of the American Mathematical Society)

   We characterize the kernel of the mixed ray transform on simple 2-dimensional Riemannian manifolds, that is, on simple surfaces for tensors of any order. 

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4. The C∞ -isomorphism property for a class of singularly-weighted x-ray transforms

   https://doi.org/10.1088/1361-6420/aca8cb  

   Mishra, Rohit Kumar; Monard, François; Zou, Yuzhou (December 2022, Inverse Problems)

   Abstract We study a one-parameter family of self-adjoint normal operators for the x-ray transform on the closed Euclidean disk D , obtained by considering specific singularly weighted L 2 topologies. We first recover the well-known singular value decompositions in terms of orthogonal disk (or generalized Zernike) polynomials, then prove that each such realization is an isomorphism of C ∞ ( D ) . As corollaries: we give some range characterizations; we show how such choices of normal operators can be expressed as functions of two distinguished differential operators. We also show that the isomorphism property also holds on a class of constant-curvature, circularly symmetric simple surfaces. These results allow to design functional contexts where normal operators built out of the x-ray transform are provably invertible, in Fréchet and Hilbert spaces encoding specific boundary behavior. 

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5. Injectivity of the Heisenberg X-ray transform

   https://doi.org/10.1016/j.jfa.2020.108886  

   Flynn, Steven (March 2021, Journal of Functional Analysis)

   We initiate the study of X-ray tomography on sub-Riemannian manifolds, for which the Heisenberg group exhibits the simplest nontrivial example. With the language of the group Fourier transform, we prove an operator-valued incarnation of the Fourier Slice Theorem, and apply this new tool to show that a sufficiently regular function on the Heisenberg group is determined by its line integrals over sub-Riemannian geodesics. We also consider the family of taming metrics $$g_\epsilon$$ approximating the sub-Riemannian metric, and show that the associated X-ray transform is injective for all $$\epsilon > 0$$. This result gives a concrete example of an injective X-ray transform in a geometry with an abundance of conjugate points. 

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