图像处理中Sobolev范数的隐正则化效应
Implicit Regularization Effects of the Sobolev Norms in Image Processing
摘要：
在本文中，我们建议使用一般的基于$L^2$的Sobolev范数，即，$H^s$范数，其中$s\in \mathbb{R}$，用于测量由图像处理任务中的噪声引起的数据差异，这些图像处理任务被公式化为最优化问题。与发展正则化方法的流行趋势相反，我们强调通过Sobolev范数类作为数据拟合项可以实现隐式正则化效果。具体来说，我们分析，隐式正则化来自权重的$H^s$规范施加在不同的频率内容的底层图像。我们进一步从贝叶斯的角度分析了使用Sobolev范数作为数据拟合项的潜在噪声假设，建立了与基于Sobolev梯度的方法的联系，并讨论了预处理对梯度下降算法收敛速度的影响，从而更好地理解函数空间/度量和图像处理中涉及的优化过程。全波反演、图像去噪和去模糊的数值结果证明了隐式正则化的效果。
英文摘要：
In this paper, we propose to use the general $L^2$-based Sobolev norms, i.e., $H^s$ norms where $s\in \mathbb{R}$, 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 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 further analyze the underlying noise assumption of using the Sobolev norm as the data-fitting term from a Bayesian perspective, build the connections with the Sobolev gradient-based methods and discuss the preconditioning effects on the convergence rate of the gradient descent algorithm, leading to a better understanding of functional spaces/metrics and the optimization process involved in image processing. Numerical results in full waveform inversion, image denoising and deblurring demonstrate the implicit regularization effects.
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