Limited data problems for the generalized Radon transform in $\mathbb{R}^n$

TL;DR

This paper investigates the generalized Radon transform in higher dimensions, providing microlocal analysis and conditions for artifact-free reconstruction using pseudodifferential filters, extending classical 2D results.

Contribution

It introduces a microlocal framework for limited data Radon transforms in $\

Findings

Characterization of visible and added singularities.

Conditions for filters to ensure ellipticity and reliable recovery.

Generalization of classical 2D Radon transform results.

Abstract

We consider the generalized Radon transform (defined in terms of smooth weight functions) on hyperplanes in $\mathbb{R}^n$. We analyze general filtered backprojection type reconstruction methods for limited data with filters given by general pseudodifferential operators. We provide microlocal characterizations of visible and added singularities in $\mathbb{R}^n$ and define modified versions of reconstruction operators that do not generate added artifacts. We calculate the symbol of our general reconstruction operators as pseudodifferential operators, and provide conditions for the filters under which the reconstruction operators are elliptic for the visible singularities. If the filters are chosen according to those conditions, we show that almost all visible singularities can be recovered reliably. Our work generalizes the results for the classical line transforms in $\mathbb{R}^2$ and the classical reconstruction operators (that use specific filters). In our proofs, we employ a general paradigm that is based on the calculus of Fourier integral operators. Since this technique does not rely on explicit expressions of the reconstruction operators, it enables us to analyze more general imaging situations.