The inverse problem for the local geodesic ray transform

By Gunther Uhlmann, András Vasy. 2012

194 citations recorded by the source corpus.

Under a convexity assumption on the boundary we solve a local inverse problem, namely we show that the geodesic X-ray transform can be inverted locally in a stable manner; one even has a reconstruction formula. We also show that under an assumption on the existence of a global foliation by strictly convex hypersurfaces the geodesic X-ray transform is globally injective. In addition we prove stability estimates and propose a layer stripping type algorithm for reconstruction.

Subfields: Biomedical Engineering, Computer Vision and Pattern Recognition, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/46b22de2adf268d099ec22173082c5f7f2bc2b6c

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In Biomedical Engineering

In Computer Vision and Pattern Recognition

In Mathematical Physics

Citations in the corpus

An inverse problem for a semi-linear elliptic equation in Riemannian geometries (2019, unknown): “Indeed, uniqueness results for the linearized problem rely on injectivity of the geodesic ray transform on (M, g) that is known to be true under strong geometric assumptions such as simplicity of the transversal manifold (M, g) or existence of a strictly convex foliation [46].”
Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge (2017, methodology): “The reason this smallness holds is that, similarly to the discussion in the scalar setting in [33], the map c 7→ N0,z+dzMδ z ∈ Ψ−1,0 sc (Xc) is continuous, meaning that if one takes a fixed space, say X0, and identifies Xc (for c small) with it via a translation, then the resulting map into Ψ−1,0 sc”
The Calderón problem in transversally anisotropic geometries (2013, background): “Calderón problem in a cylinder: continuous spectrum 38 References 46”