Uncertainty Quantification in Imaging: When Convex Optimization Meets Bayesian Analysis

We propose to perform Bayesian uncertainty quantification via convex optimization tools (BUQO), in the context of high dimensional inverse problems. We quantify the uncertainty associated with particular structures appearing in the maximum a posteriori estimate, obtained from a log-concave Bayesian model. A hypothesis test is defined, where the null hypothesis represents the non-existence of the structure of interest in the true image. To determine if this null hypothesis is rejected, we use the data and prior knowledge. Computing such test in the context of imaging problem is often intractable due to the high dimensionality involved. In this work, we propose to leverage probability concentration phenomena and the underlying convex geometry to formulate the Bayesian hypothesis test as a convex minimization problem. This problem is subsequently solved using a proximal primal-dual algorithm. The proposed method is applied to astronomical radio-interferometric imaging.