  Scientific imaging problems are often severely ill-posed, and hence have
significant intrinsic uncertainty. Accurately quantifying the uncertainty in
the solutions to such problems is therefore critical for the rigorous
interpretation of experimental results as well as for reliably using the
reconstructed images as scientific evidence. Unfortunately, existing imaging
methods are unable to quantify the uncertainty in the reconstructed images in a
manner that is robust to experiment replications. This paper presents a new
uncertainty quantification methodology based on an equivariant formulation of
the parametric bootstrap algorithm that leverages symmetries and invariance
properties commonly encountered in imaging problems. Additionally, the proposed
methodology is general and can be easily applied with any image reconstruction
technique, including unsupervised training strategies that can be trained from
observed data alone, thus enabling uncertainty quantification in situations
where there is no ground truth data available. We demonstrate the proposed
approach with a series of numerical experiments and through comparisons with
alternative uncertainty quantification strategies from the state-of-the-art,
such as Bayesian strategies involving score-based diffusion models and Langevin
samplers. In all our experiments, the proposed method delivers remarkably
accurate high-dimensional confidence regions and outperforms the competing
approaches in terms of estimation accuracy, uncertainty quantification
accuracy, and computing time.
