Optimal control of the fidelity coefficient in a Cahn-Hilliard image inpainting model
Elena Beretta, Cecilia Cavaterra, Matteo Fornoni, Maurizio Grasselli
TL;DR
The paper addresses optimizing the fidelity coefficient in a Cahn–Hilliard–type image inpainting model by formulating a controlled evolution with a phase field $\varphi \in [-1,1]$ and a logarithmic-type potential to ensure boundedness. It proves well-posedness of the state system, existence of an optimal control, and derives first-order optimality conditions via an adjoint system and a linearised state equation; it further establishes second-order sufficient conditions for local optimality using a control-to-costate framework. The results provide a rigorous foundation for spatially varying fidelity in PDE-based inpainting, enabling principled tuning of data fidelity while penalizing small fidelity where needed. The framework is poised for numerical implementation and extensions to color images, anisotropic regularisation, and time–space adaptive fidelity strategies, with potential impact on image restoration quality and computational efficiency.
Abstract
We consider an inpainting model proposed by A. Bertozzi et al., which is based on a Cahn-Hilliard-type equation. This equation describes the evolution of an order parameter that represents an approximation of the original image occupying a bounded two-dimensional domain. The given image is assumed to be damaged in a fixed subdomain, and the equation is characterised by a linear reaction term. This term is multiplied by the so-called fidelity coefficient, which is a strictly positive bounded function defined in the undamaged region. The idea is that, given an initial image, the order parameter evolves towards the given image, and this process properly diffuses through the boundary of the damaged region, restoring the damaged image, provided that the fidelity coefficient is large enough. Here, we formulate an optimal control problem based on this fact, namely, our cost functional accounts for the magnitude of the fidelity coefficient. Assuming a singular potential to ensure that the order parameter takes its values in between 0 and 1, we first analyse the control-to-state operator and prove the existence of at least one optimal control, establishing the validity of first-order optimality conditions. Then, under suitable assumptions, we demonstrate second-order optimality conditions.