Sampling the X-ray transform on simple surfaces

2021/10/12 by François Monard, Plamen Stefanov, Monard, François +1 · 2 citations
Engineering · Mathematics · Medicine · #3D Shape Modeling and Analysis #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Medical Imaging Techniques and Applications #Morphological variations and asymmetry

paper · pdf · doi:10.48550/arxiv.2110.05761

openalex publication_date 2021/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Cited by

• A family of non-simple surfaces whose transport twistor spaces admit global blow-down maps
• Sampling theorems for inverse problems on Riemannian manifolds

Related

• The attenuated geodesic X-ray transform
• Geodesic X-ray transform and streaking artifacts on simple surfaces or on spaces of constant curvature
• Reconstruction formulas for X-ray transforms in negative curvature
• Inversion of the attenuated geodesic X-ray transform over functions and vector fields on simple surfaces
• Functional relations, sharp mapping properties and regularization of the X-ray transform on disks of constant curvature