# Sampling the X-ray transform on simple surfaces

@inproceedings{Monard2021SamplingTX, title={Sampling the X-ray transform on simple surfaces}, author={Franccois Monard and Plamen Stefanov}, year={2021} }

We study the problem of proper discretizing and sampling issues related to geodesic X-ray transforms on simple surfaces, and illustrate the theory on simple geodesic disks of constant curvature. Given a notion of band limit on a function, we provide the minimal sampling rates of its X-ray transform for a faithful reconstruction. In Cartesian sampling, we quantify the quality of a sampling scheme depending on geometric parameters of the surface (e.g. curvature and boundary curvature), and the…

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SHOWING 1-10 OF 20 REFERENCES

The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points

- Physics, Mathematics
- 2014

We study the geodesic X-ray transform X on compact Riemannian surfaces with conjugate points. Regardless of the type of the conjugate points, we show that we cannot recover the singularities and,…

Inversion of the Attenuated Geodesic X-Ray Transform over Functions and Vector Fields on Simple Surfaces

- Mathematics, Computer ScienceSIAM J. Math. Anal.
- 2016

We derive explicit reconstruction formulas for the attenuated geodesic X-ray transform over functions and, in the case of nonvanishing attenuation, vector fields, on a class of simple Riemannian…

On characterization of the range and inversion formulas for the geodesic X-ray transform

- Mathematics
- 2004

We describe a relation between the scattering relation, the Hilbert transform in frequency space, and the geodesic ray transform for simple, two-dimensional compact Riemannian manifolds with…

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

- Mathematics
- 2015

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…

Ray Transform on Riemannian Manifolds

- Mathematics
- 2004

What is integral geometry? Since the famous paper by I. Radon in 1917, it has been agreed that integral geometry problems consist in determining sonic function or a more general object (cohomology…

Sharp stability estimate for the geodesic ray transform

- Mathematics
- 2018

We prove a sharp $L^2\to H^{1/2}$ stability estimate for the geodesic X-ray transform of tensor fields of order $0$, $1$ and $2$ on a simple Riemannian manifold with a suitable chosen $H^{1/2}$ norm.…

Range characterizations and Singular Value Decomposition of the geodesic X-ray transform on disks of constant curvature

- Physics, MathematicsJournal of Spectral Theory
- 2021

For a one-parameter family of simple metrics of constant curvature ($4\kappa$ for $\kappa\in (-1,1)$) on the unit disk $M$, we first make explicit the Pestov-Uhlmann range characterization of the…

Integral Geometry of Tensor Fields

- Mathematics
- 1994

Introduction: the problem of determining a metric by its hodograph and a linearization of the problem the kinetic equation in a Riemannian manifold. Part 1 The ray transform of symmetric tensor…

Semiclassical Sampling and Discretization of Certain Linear Inverse Problems

- Computer Science, MathematicsSIAM J. Math. Anal.
- 2020

A Weyl type of estimate is proved on the minimal number of sampling points to recover $f$ stably in terms of the volume of its semiclassical wave front set.

Sampling in Fan Beam Tomography

- Mathematics, Computer ScienceSIAM J. Appl. Math.
- 1993

Using a sampling theorem for periodic functions and asymptotic estimates for the Fourier transform of the fan beam transform, the exact sampling conditions are found for standard fan beam scanning necessary to obtain a certain resolution.