The geodesic X-ray transform with matrix weights
Paternain, G. B., Salo, M., Uhlmann, G., & Zhou, H. (2019). The geodesic X-ray transform with matrix weights. American Journal of Mathematics, 141(6), 1707-1750. https://doi.org/10.1353/ajm.2019.0045
Published in
American Journal of MathematicsDate
2019Discipline
Inversio-ongelmien huippuyksikköMatematiikkaCentre of Excellence in Inverse ProblemsMathematicsCopyright
© 2019 by Johns Hopkins University Press
Consider a compact Riemannian manifold of dimension ≥ 3 with strictly convex boundary, such that the manifold admits a strictly convex function. We show that the attenuated ray transform in the presence of an arbitrary connection and Higgs field is injective modulo the natural obstruction for functions and one-forms. We also show that the connection and the Higgs field are uniquely determined by the scattering relation modulo gauge transformations. The proofs involve a reduction to a local result showing that the geodesic X-ray transform with a matrix weight can be inverted locally near a point of strict convexity at the boundary, and a detailed analysis of layer stripping arguments based on strictly convex exhaustion functions. As a somewhat striking corollary, we show that these integral geometry problems can be solved on strictly convex manifolds of dimension ≥ 3 having nonnegative sectional curvature (similar results were known earlier in negative sectional curvature). We also apply our methods to solve some inverse problems in quantum state tomography and polarization tomography.
...


Publisher
Johns Hopkins University PressISSN Search the Publication Forum
0002-9327Keywords
Publication in research information system
https://converis.jyu.fi/converis/portal/detail/Publication/34636999
Metadata
Show full item recordCollections
Related funder(s)
European Commission; Academy of FinlandFunding program(s)
FP7 (EU's 7th Framework Programme); Centre of Excellence, AoF; Academy Project, AoF


The content of the publication reflects only the author’s view. The funder is not responsible for any use that may be made of the information it contains.
Additional information about funding
Research of the first and fourth authors supported by EPSRC grant EP/M023842/1; research of the second author supported by the Academy of Finland (Finnish Centre of Excellence in Inverse Modelling and Imaging, grant numbers 284715 and 309963) and by the European Research Council under FP7/2007-2013 (ERC StG 307023) and Horizon 2020 (ERC CoG 770924); research of the third author supported in part by the NSF.

License
Related items
Showing items with similar title or keywords.
-
Geodesic ray transform with matrix weights for piecewise constant functions
Ilmavirta, Joonas; Railo, Jesse (Suomalainen tiedeakatemia, 2020)We show injectivity of the geodesic X-ray transform on piecewise constant functions when the transform is weighted by a continuous matrix weight. The manifold is assumed to be compact and nontrapping of any dimension, and ... -
Unique Continuation Results for Certain Generalized Ray Transforms of Symmetric Tensor Fields
Agrawal, Divyansh; Krishnan, Venkateswaran P.; Sahoo, Suman Kumar (Springer Science and Business Media LLC, 2022)Let Im denote the Euclidean ray transform acting on compactly supported symmetric m-tensor field distributions f, and I∗m be its formal L2 adjoint. We study a unique continuation result for the operator Nm=I∗mIm. More ... -
Limiting Carleman weights and conformally transversally anisotropic manifolds
Angulo, Pablo; Faraco, Daniel; Guijarro, Luis; Salo, Mikko (American Mathematical Society, 2020)We analyze the structure of the set of limiting Carleman weights in all conformally flat manifolds, $ 3$-manifolds, and $ 4$-manifolds. In particular we give a new proof of the classification of Euclidean limiting Carleman ... -
X-ray Transforms in Pseudo-Riemannian Geometry
Ilmavirta, Joonas (Springer US, 2018)We study the problem of recovering a function on a pseudo-Riemannian manifold from its integrals over all null geodesics in three geometries: pseudo-Riemannian products of Riemannian manifolds, Minkowski spaces, and tori. ... -
The Geodesic Ray Transform on Spherically Symmetric Reversible Finsler Manifolds
Ilmavirta, Joonas; Mönkkönen, Keijo (Springer Science and Business Media LLC, 2023)We show that the geodesic ray transform is injective on scalar functions on spherically symmetric reversible Finsler manifolds where the Finsler norm satisfies a Herglotz condition. We use angular Fourier series to reduce ...