Bilevel Programming Approach for Image Restoration Problems with Automatically Hyperparameter Selection

In optimization-based image restoration models, the correct selection of hyperparameters is crucial for achieving superior performance. However, current research typically involves manual tuning of these hyperparameters, which is highly time-consuming and often lacks accuracy. In this paper, we concentrate on the automated selection of hyperparameters in the context of image restoration and present a bilevel programming approach that can simultaneously select the optimal hyperparameters and achieve high-quality restoration results. For implementation, we reformulate the bilevel programming problem that incorporates an inequality constraint related to the difference-of-convex functions. Following this, we address a sequence of nonsmooth convex programming problems by employing a feasibility penalty function along with a proximal point term. In this context, the nonsmooth convex programming problem uses the solution of the lower-level problem, which is derived through the alternating direction method of multipliers. Theoretically, we prove that the sequence generated by the algorithm converges to a Karush-Kuhn-Tucker stationary point of the inequality-constrained equivalent bilevel programming problem. We conduct a series of tests on both simulations and real images, which demonstrate that the proposed algorithm achieve superior restoration quality while requiring less computing time compared to other hyperparameter selection methods.