<p style='text-indent:20px;'>In this paper, we focus on learning optimal parameters for PDE-based image denoising and decomposition models. First, we learn the regularization parameter and the differential operator for gray-scale image denoising using the fractional Laplacian in combination with a bilevel optimization problem. In our setting the fractional Laplacian allows the use of Fourier transform, which enables the optimization of the denoising operator. We prove stable and explainable results as an advantage in comparison to machine learning approaches. The numerical experiments correlate with our theoretical model settings and show a reduction of computing time in contrast to the Rudin-Osher-Fatemi model. Second, we introduce a new regularized image decomposition model with the fractional Laplacian and the Riesz potential. We provide an explicit formula for the unique solution and the numerical experiments illustrate the efficiency.</p>

中文翻译：

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参数学习和分数微分算子：正则化图像去噪和分解问题中的应用

<p style='text-indent:20px;'>在本文中，我们专注于学习基于 PDE 的图像去噪和分解模型的最优参数。首先，我们使用分数拉普拉斯算子结合双层优化问题来学习灰度图像去噪的正则化参数和微分算子。在我们的设置中，分数拉普拉斯算子允许使用傅里叶变换，这可以优化去噪算子。与机器学习方法相比，我们证明了稳定且可解释的结果是一个优势。数值实验与我们的理论模型设置相关，并显示与 Rudin-Osher-Fatemi 模型相比计算时间减少。其次，我们引入了一种新的具有分数拉普拉斯和 Riesz 势的正则化图像分解模型。