Confidence regions for images observed under the Radon transform

Nicolai Bissantz, Hajo Holzmann, Katharina Proksch

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Abstract

Recovering a function ƒ from its integrals over hyperplanes (or line integrals in the two-dimensional case), that is, recovering ƒ from the Radon transform Rƒ of ƒ, is a basic problem with important applications in medical imaging such as computerized tomography (CT). In the presence of stochastic noise in the observed function Rƒ, we shall construct asymptotic uniform confidence regions for the function ƒ of interest, which allows to draw conclusions regarding global features of ƒ. Specifically, in a white noise model as well as a fixed-design regression model, we prove a Bickel–Rosenblatt-type theorem for the maximal deviation of a kernel-type estimator from its mean, and give uniform estimates for the bias for ƒ in a Sobolev smoothness class. The finite sample properties of the proposed methods are investigated in a simulation study.
Original languageEnglish
Pages (from-to)86-107
JournalJournal of multivariate analysis
Volume128
DOIs
Publication statusPublished - Jul 2014
Externally publishedYes

Keywords

  • Confidence bands
  • Inverse problems
  • Nonparametric regression
  • Radon transform

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