A Scaled Gradient Projection method for the realization of the balancing principle in TGV-based image restoration

In the last few years, Total Generalized Variation (TGV) regularization has proved to be a valuable tool to remove blur and noise from an image while avoiding the staircase effect typical of the Total Variation (TV) and preserving the sharp edges. The TGV-regularized model depends on two regularization parameters whose values must be appropriately selected to obtain good-quality restored images. In this work, we propose the use of the Balancing Principle (BP) to formulate the TGV-based image restoration problem as a constrained minimization problem whose objective is an implicit function of the two regularization parameters depending on the image to be restored. The values of the regularization parameters, and the corresponding restored image, satisfying the optimality condition of the formulated problem guarantee that the data fidelity and regularization terms are balanced. We introduce a Scaled Gradient Projection (SGP) method specifically tailored to the BP-based optimization problem and test its effectiveness against the fixed-point iteration schemes proposed in the literature. The numerical results performed on real-life images, affected by both Gaussian and Poisson noise, show that the proposed approach can effectively restore input images corrupted by several kinds of noise and outperform the fixed-point strategies for the realization of the Balancing Principle.