Opuscula Math. 40, no. 3 (2020), 361-373
https://doi.org/10.7494/OpMath.2020.40.3.361

Opuscula Mathematica

# A note on confidence intervals for deblurred images

Michał Biel
Zbigniew Szkutnik

Abstract. We consider pointwise asymptotic confidence intervals for images that are blurred and observed in additive white noise. This amounts to solving a stochastic inverse problem with a convolution operator. Under suitably modified assumptions, we fill some apparent gaps in the proofs published in [N. Bissantz, M. Birke, Asymptotic normality and confidence intervals for inverse regression models with convolution-type operators, J. Multivariate Anal. 100 (2009), 2364-2375]. In particular, this leads to modified bootstrap confidence intervals with much better finite-sample behaviour than the original ones, the validity of which is, in our opinion, questionable. Some simulation results that support our claims and illustrate the behaviour of the confidence intervals are also presented.

Keywords: inverse problems, confidence intervals, convolution, deblurring.

Mathematics Subject Classification: 62G08, 62G15, 62G20.

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• Michał Biel
• Faculty of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Krakow, Poland
• Zbigniew Szkutnik (corresponding author)
• https://orcid.org/0000-0002-4607-6268
• Faculty of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Krakow, Poland
• Communicated by Mirosław Pawlak.
• Revised: 2019-11-12.
• Accepted: 2020-02-04.
• Published online: 2020-04-04.