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.

中文翻译：

---

图像处理中 Sobolev 范数的隐式正则化效果

在本文中，我们建议使用一般基于 $L^2$ 的 Sobolev 范数（即 $H^s$ 范数，$s\in \mathbb{R}$）来衡量由于图像中的噪声引起的数据差异处理被公式化为优化问题的任务。与开发正则化方法的流行趋势相反，我们强调可以通过将 Sobolev 范数作为数据拟合项来实现 \textit{implicit} 正则化效果。具体来说，我们分析了隐式正则化来自 $H^s$ 范数对底层图像的不同频率内容施加的权重。我们还将此类规范与基于最佳传输的度量和基于 Sobolev 梯度的方法建立了联系，从而更好地理解功能空间/度量以及图像处理中涉及的优化过程。我们使用快速傅立叶变换有效地计算$H^s$范数，并将其与乘法器交替方向法（ADMM）框架中的总变化正则化相结合。去噪和去模糊的数值结果支持我们的理论发现。