Maximum likelihood estimation of regularization parameters in high-dimensional inverse problems: An empirical bayesian approach part i: Methodology and experiments

Many imaging problems require solving an inverse problem that is ill-conditioned or ill-posed. Imaging methods typically address this difficulty by regularizing the estimation problem to make it well-posed. This often requires setting the value of the so-called regularization parameters that control the amount of regularization enforced. These parameters are notoriously difficult to set a priori and can have a dramatic impact on the recovered estimates. In this work, we propose a general empirical Bayesian method for setting regularization parameters in imaging problems that are convex w.r.t. the unknown image. Our method calibrates regularization parameters directly from the observed data by maximum marginal likelihood estimation and can simultaneously estimate multiple regular-ization parameters. Furthermore, the proposed algorithm uses the same basic operators as proximal optimization algorithms, namely gradient and proximal operators, and it is therefore straightfor-ward to apply to problems that are currently solved by using proximal optimization techniques. Our methodology is demonstrated with a range of experiments and comparisons with alternative approaches from the literature. The considered experiments include image denoising, nonblind image deconvolution, and hyperspectral unmixing, using synthesis and analysis priors involving the ℓ₁, total-variation, total-variation and ℓ₁, and total-generalized-variation pseudonorms. A detailed theoretical analysis of the proposed method is presented in our companion paper.