Sampling the X-ray transform on simple surfaces

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 coordinate system used to represent the space of geodesics. When aliasing happens, we explain how to predict the location, orientation and frequency of the artifacts.

[1]  F. Natterer,et al.  Sampling in Fan Beam Tomography , 1993, SIAM J. Appl. Math..

[2]  Semiclassical analysis , 2019, Graduate Studies in Mathematics.

[3]  François Monard,et al.  Inversion of the Attenuated Geodesic X-Ray Transform over Functions and Vector Fields on Simple Surfaces , 2015, SIAM J. Math. Anal..

[4]  G. Uhlmann,et al.  On the microlocal analysis of the geodesic X-ray transform with conjugate points , 2015, 1502.06545.

[5]  Vladimir A. Sharafutdinov,et al.  Ray Transform on Riemannian Manifolds , 2004 .

[6]  S. Tindel,et al.  Sampling linear inverse problems with noise. , 2020, 2011.13489.

[7]  Rohit Kumar Mishra,et al.  Range characterizations and Singular Value Decomposition of the geodesic X-ray transform on disks of constant curvature , 2021, Journal of Spectral Theory.

[8]  André Martinez,et al.  An Introduction to Semiclassical and Microlocal Analysis , 2002 .

[9]  David Middleton,et al.  Sampling and Reconstruction of Wave-Number-Limited Functions in N-Dimensional Euclidean Spaces , 1962, Inf. Control..

[10]  Chase Mathison.  Sampling in thermoacoustic tomography , 2019, ArXiv.

[11]  V. Sharafutdinov.  Integral Geometry of Tensor Fields , 1994 .

[12]  Yernat M. Assylbekov,et al.  Sharp stability estimate for the geodesic ray transform , 2018, Inverse Problems.

[13]  Joonas Ilmavirta,et al.  4. Integral geometry on manifolds with boundary and applications , 2018, The Radon Transform.

[14]  G. Uhlmann,et al.  The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points , 2014, 1402.5559.

[15]  Plamen Stefanov,et al.  Semiclassical Sampling and Discretization of Certain Linear Inverse Problems , 2018, SIAM J. Math. Anal..

[16]  G. Uhlmann,et al.  On characterization of the range and inversion formulas for the geodesic X-ray transform , 2004 .