Posterior-Variance-Based Error Quantification for Inverse Problems in Imaging

DOI10.1137/23M1546129arXiv2212.12499OpenAlexW4391613146WikidataQ128331800 ScholiaQ128331800MaRDI QIDQ6421566

Thomas Pock, Andreas Habring, Dominik Narnhofer, Martin Holler

Publication date: 23 December 2022

Abstract: In this work, a method for obtaining pixel-wise error bounds in Bayesian regularization of inverse imaging problems is introduced. The proposed method employs estimates of the posterior variance together with techniques from conformal prediction in order to obtain coverage guarantees for the error bounds, without making any assumption on the underlying data distribution. It is generally applicable to Bayesian regularization approaches, independent, e.g., of the concrete choice of the prior. Furthermore, the coverage guarantees can also be obtained in case only approximate sampling from the posterior is possible. With this in particular, the proposed framework is able to incorporate any learned prior in a black-box manner. Guaranteed coverage without assumptions on the underlying distributions is only achievable since the magnitude of the error bounds is, in general, unknown in advance. Nevertheless, experiments with multiple regularization approaches presented in the paper confirm that in practice, the obtained error bounds are rather tight. For realizing the numerical experiments, also a novel primal-dual Langevin algorithm for sampling from non-smooth distributions is introduced in this work.

Full work available at URL: https://doi.org/10.1137/23m1546129

Mathematics Subject Classification ID

Bayesian inference (62F15) Computing methodologies for image processing (68U10) Image processing (compression, reconstruction, etc.) in information and communication theory (94A08) Numerical solution to inverse problems in abstract spaces (65J22)

Related Items (3)

Neural-network-based regularization methods for inverse problems in imaging ⋮ Subgradient Langevin methods for sampling from nonsmooth potentials ⋮ Robustness and exploration of variational and machine learning approaches to inverse problems: an overview

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