Robust Image Restoration with an Adaptive Huber Function Based Fidelity

Abstract

Numerous image restoration algorithms have been proposed in the last several decades. These algorithms usually optimize an objective function consisting of an ${\ell _2}$ norm based fidelity and a regularization term, whose optimality could be justified from the view of maximum a posteriori estimation with an assumption that the noise is Gaussian. However, it is known that the ${\ell _2}$ norm based fidelity is very sensitive to gross errors that may appear in the observation. Since real-world image restoration tasks are usually hindered by abnormal pixels, impulsive noise, and other heavy-tailed noise, the utility of these traditional algorithms is limited. Although some robust algorithms have been proposed by replacing the ${\ell _2}$ norm based fidelity with a robust one, they are designed for specific restoration tasks (e.g., multi-frame super-resolution) with a fixed image prior (e.g., the total-variation) and have not provided a principled way to justify the choice of a robust fidelity term. Currently designing a robust algorithm for general image restoration tasks is still an open problem. This paper studies the problem of robust image restoration in both theoretical and algorithmic manners. In the theoretical part, we point out that Huber function based fidelity could be justified from the pespective of minimax estimation, which facilities the choice of the robust fidelity term. In the algorithmic part, we first propose an adaptive approach to set the threshold of the Huber function, and then we derive an efficient and flexible method to solve the proposed robust formulation of the image restoration problem, which enables the proposed algorithm to incorporate various image priors. Experiments have demonstrated the robustness of the proposed algorithm and its utility in real-world image restoration tasks.