A general framework for whiteness-based parameters selection in variational models

In this work, we extend the residual whiteness principle, originally proposed in (Lanza et al. in Electron Trans Numer Anal 53:329–352 2020) for the selection of a single regularization parameter in variational models for inverse problems under additive white noise corruption, to much broader scenarios. More specifically, we address the problem of estimating multiple parameters for imaging inverse problems subject to both white and non-white but whitenable noise corruptions, thus covering most of the application cases. The proposed parameter selection criterion, referred to as generalized whiteness principle, is formulated as a bilevel optimization problem. To circumvent the non-smoothness of the variational models typically employed in imaging problems—the non-smoothness representing a bottleneck in the bilevel set-up—we propose to adopt a derivative-free minimization algorithm for the solution of the designed bilevel problem. We refer to this novel numerical solution paradigm as bilevel derivative-free approach. Numerical tests highlight both the ability of the proposed generalized whiteness principle to effectively select multiple parameters and the significant advantages, in terms of computational cost, of the bilevel derivative-free numerical solution framework.