A Direct Sampling Method for the Inversion of the Radon Transform

TL;DR

This paper introduces a new direct sampling method for Radon transform inversion that enhances robustness and computational efficiency through fractional Sobolev duality and novel probing functions.

Contribution

The paper develops a generalized DSM using fractional Sobolev duality and new probing functions, improving stability and speed in Radon transform inversion.

Findings

Enhanced robustness in ill-posed inverse problems

Fast and parallelizable computation of the DSM

Numerical experiments show improved accuracy and stability

Abstract

We propose a novel direct sampling method (DSM) for the effective and stable inversion of the Radon transform. The DSM is based on a generalization of the important almost orthogonality property in classical DSMs to fractional order Sobolev duality products and to a new family of probing functions. The fractional order duality product proves to be able to greatly enhance the robustness of the reconstructions in some practically important but severely ill-posed inverse problems associated with the Radon transform. We present a detailed analysis to better understand the performance of the new probing and index functions, which are crucial to stable and effective numerical reconstructions. The DSM can be computed in a very fast and highly parallel manner. Numerical experiments are carried out to compare the DSM with a popular existing method, and to illustrate the efficiency, stability, and accuracy of the DSM.