Implicit Regularization Effects of the Sobolev Norms in Image Processing
Implicit Regularization Effects of the Sobolev Norms in Image Processing
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DOI：
10.1007/s44007-023-00077-8
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发表时间：
2021-09
期刊：
影响因子：
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通讯作者：
Yunan Yang;Jingwei Hu;Y. Lou
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文献类型：
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作者：
Yunan Yang;Jingwei Hu;Y. Lou
在本文中，我们建议使用基于一般的Sobolev范数，即，其中，用于测量由于图像处理任务中的噪声而导致的数据差异，这些图像处理任务被公式化为优化问题。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。具体地说，我们分析了隐式正则化来自权值，thebrum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。在图像去噪和全波形反演两个地球物理应用中的数值结果证明了隐式正则化的效果。
In this paper, we propose to use the general-based Sobolev norms, i.e.,norms where, 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 animplicitregularization 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 thenorm 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 two geophysical applications of image denoising and full waveform inversion demonstrate the implicit regularization effects.