Asymptotics of the mean-square risk estimate in the problem of inverting the Radon transform from projections registered on a random grid

O. V. Shestakov<sup>ab

a</sup> Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, M. V. Lomonosov Moscow State University, 1-52 Leninskiye Gory, GSP-1, Moscow 119991, Russian Federation
^b Institute of Informatics Problems, Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 44-2 Vavilov Str., Moscow 119333, Russian Federation

Abstract: When reconstructing tomographic images, it is necessary to use regularization methods, since the problem of inverting the Radon transform, which is the basis of mathematical models of most tomographic experiments, is ill-posed. Regularization methods based on wavelet analysis have become popular due to their adaptation to local image features and computational efficiency. The analysis of errors in tomographic images is an important practical task, since it makes it possible to evaluate the quality of both the methods themselves and the equipment used. Sometimes, it is not possible to register projection data on a uniform grid of samples. If sample points for each coordinate form a variation series based on a sample from a uniform distribution, then the use of the usual threshold processing procedure is adequate. In this paper, the author analyzes the estimate of the mean-square risk in the Radon transform inversion problem and demonstrates that if the image function is uniformly Lipschitz-regular, then this estimate is strongly consistent and asymptotically normal.

Keywords: threshold processing, Radon transform, random grid, mean-square risk estimate.

Received: 07.04.2020

DOI: 10.14357/19922264200204