Implicit Regularization Effects of the Sobolev Norms in Image Processing

In this paper, we propose to use the general $L^2$-based Sobolev norms (i.e., $H^s$ norms, $s\in \mathbb{R}$) to measure the data discrepancy due to noise in image processing tasks that are formulated as optimization problems. As opposed to a popular trend of developing regularization methods, we emphasize that an \textit{implicit} regularization effect can be achieved through the class of Sobolev norms as the data-fitting term. Specifically, we analyze that the implicit regularization comes from the weights that the $H^s$ norm imposes on different frequency contents of an underlying image. We also build the connections of such norms with the optimal transport-based metrics and the Sobolev gradient-based methods, leading to a better understanding of functional spaces/metrics and the optimization process involved in image processing. We use the fast Fourier transform to compute the $H^s$ norm efficiently and combine it with the total variation regularization in the framework of the alternating direction method of multipliers (ADMM). Numerical results in both denoising and deblurring support our theoretical findings.