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.
- Confidence bands
- Inverse problems
- Nonparametric regression
- Radon transform
Bissantz, N., Holzmann, H., & Proksch, K. (2014). Confidence regions for images observed under the Radon transform. Journal of multivariate analysis, 128, 86-107. https://doi.org/10.1016/j.jmva.2014.03.005