Confidence regions for images observed under the Radon transform

Nicolai Bissantz, Hajo Holzmann, Katharina Proksch

Research output: Contribution to journalArticleAcademicpeer-review

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

Cite this

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title = "Confidence regions for images observed under the Radon transform",
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.",
keywords = "Confidence bands, Inverse problems, Nonparametric regression, Radon transform",
author = "Nicolai Bissantz and Hajo Holzmann and Katharina Proksch",
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language = "English",
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Confidence regions for images observed under the Radon transform. / Bissantz, Nicolai; Holzmann, Hajo; Proksch, Katharina.

In: Journal of multivariate analysis, Vol. 128, 07.2014, p. 86-107.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Confidence regions for images observed under the Radon transform

AU - Bissantz, Nicolai

AU - Holzmann, Hajo

AU - Proksch, Katharina

PY - 2014/7

Y1 - 2014/7

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AB - 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.

KW - Confidence bands

KW - Inverse problems

KW - Nonparametric regression

KW - Radon transform

U2 - 10.1016/j.jmva.2014.03.005

DO - 10.1016/j.jmva.2014.03.005

M3 - Article

VL - 128

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EP - 107

JO - Journal of multivariate analysis

JF - Journal of multivariate analysis

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