Uncertainty Quantification in Astro-Imaging by Optimisation

In our recent work [1], we proposed a Bayesian uncertainty quantification method for large scale imaging inverse problems. It aims to analyse the confidence in specific structures observed in maximum a posteriori (MAP) estimates (e.g., lesions in medical imaging, celestial sources in astronomical imaging), assuming a log-concave Bayesian model. This information can subsequently be used as evidence to inform decisions and conclusions. We propose to perform a hypothesis test on the structures of interest, to assert their uncertainty. The test is formulated as a convex minimisation problem, enabling the use of advanced optimisation algorithms. In this abstract we summarise the proposed Bayesian Uncertainty Quantification by Optimisation (BUQO) method and we provide results obtained on real data in the context of radio-astronomical imaging. I. IMAGING INVERSE PROBLEMS In many imaging inverse problems, the objective is to find an estimate x† ∈ R of an original unknown image x ∈ R from a degraded observation y = Φx + w, where Φ ∈ CM×N is the measurement linear operator and w ∈ C is a realization of a random noise. In this work, we assume that the noise has a bounded energy per block, i.e. there exist (εd)1≤d≤D ∈]0,+∞[ such that ‖wd‖2 ≤ εd, wd ∈ Cd , M = M1 + . . . + Md, and w = (wd)1≤d≤D . The forward model can then be rewritten as yd = Φdx + wd, where Φd ∈ CMd×N is a subpart of Φ mapping x to yd ∈ Cd . A common Bayesian approach consists in modelling x as a random vector with prior distribution p(x) that is log-concave, and using a MAP estimation given by x† ∈ Argmin fy +g, where fy(x) = − log p(y|x) is associated with the forward model and g(x) = − log p(x). In particular, we use fy(x) = ∑D d=1 ιB2(yd,εd)(Φdx). and g(x) = ι[0,+∞[N (x)+λ‖Ψx‖1, where Ψ ∈ RL×N is a sparsity basis and λ > 0. The MAP approach provides a single point estimate that can be easily visually analysed. Nevertheless, in many applications it is necessary to analyse as well the uncertainty in the delivered solutions. We propose to address this point by leveraging probability concentration phenomena and the model’s underlying convex geometry to formulate Bayesian hypothesis tests as convex minimisation problems.