Related papers: Reconstruction of piecewise constant functions fro…

We show that on a two-dimensional compact nontrapping manifold with strictly convex boundary, a piecewise constant function is determined by its integrals over geodesics. In higher dimensions, we obtain a similar result if the manifold…

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 in dimension three…

In this paper a compact Riemannian manifold with strictly convex boundary is reconstructed from its partial travel time data. This data assumes that an open measurement region on the boundary is given, and that for every point in the…

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…

This short note modifies a reconstruction method by the author (Comm. PDE, 45(9):1118-1133, 2020), for reconstructing piecewise constant conductivities in the Calder\'on problem (electrical impedance tomography). In the former paper, a…

Given a smooth non-trapping compact manifold with strictly con- vex boundary, we consider an inverse problem of reconstructing the manifold from the scattering data initiated from internal sources. This data consist of the exit directions…

We consider the recovery of a continuous domain piecewise constant image from its non-uniform Fourier samples using a convex matrix completion algorithm. We assume the discontinuities/edges of the image are localized to the zero levelset of…

We consider the problem of developing a method to reconstruct a potential $q$ from the partial data Dirichlet-to-Neumann map for the Schr\"odinger equation $(-\Delta_g+q)u=0$ on a fixed admissible manifold $(M,g)$. If the part of the…

In this short note, we prove that all geodesically convex functions defined on a Riemannian manifold are continuous in the interior of their domain. This is a folklore result, but to the best of our knowledge, there is only one available…

We derive reconstruction formulas for a family of geodesic ray transforms with connection, defined on simple Riemannian surfaces. Such formulas provide injectivity of such all transforms in a neighbourhood of constant curvature metrics and…

We derive explicit reconstruction formulas for the attenuated geodesic X-ray transform over functions and, in the case of non-vanishing attenuation, vector fields, on a class of simple Riemannian surfaces with boundary. These formulas…

We consider the problem of recovering a compactly-supported function from a finite collection of pointwise samples of its Fourier transform taking nonuniformly. First, we show that under suitable conditions on the sampling frequencies -…

From a sufficiently large point sample lying on a compact Riemannian submanifold of Euclidean space, one can construct a simplicial complex which is homotopy-equivalent to that manifold with high confidence. We describe a corresponding…

We study the geodesic X-ray transform $I_\Gamma$ of tensor fields on a compact Riemannian manifold $M$ with non-necessarily convex boundary and with possible conjugate points. We assume that $I_\Gamma$ is known for geodesics belonging to an…

It is shown that Paley-Wiener functions on Riemannian manifolds of bounded geometry can be reconstructed in a stable way from some countable sets of their inner products with certain distributions of compact support. A reconstruction method…

We present a numerical implementation of the geodesic ray transform and its inversion over functions and solenoidal vector fields on two-dimensional Riemannian manifolds. For each problem, inversion formulas previously derived in…

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. We give proofs of…

We investigate the problem of reconstructing a 2D piecewise smooth function from its bandlimited Fourier measurements. This is a well known and well studied problem with many real world implications, in particular in medical imaging. While…

We prove the local invertibility, up to potential fields, and stability of the geodesic X-ray transform on tensor fields of order 1 and 2 near a strictly convex boundary point, on manifolds with boundary of dimension n>=3. We also present…

We consider the proximal gradient method on Riemannian manifolds for functions that are possibly not geodesically convex. Starting from the forward-backward-splitting, we define an intrinsic variant of the proximal gradient method that uses…