论文标题
参数学习和分数差分运算符：在图像正则化和分解中应用
Parameter learning and fractional differential operators: application in image regularization and decomposition
论文作者
论文摘要
在本文中，我们专注于学习基于PDE的图像正则化和分解的最佳参数。首先，我们了解使用分数拉普拉斯（Laplacian）与双重优化问题结合使用的灰度图像denoising的正则化参数和差分运算符。在我们的设置中，分数laplacian允许使用傅立叶变换，从而可以优化去核操作员。与其他机器学习方法相比，我们证明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，并且与ROF模型相比显示了计算时间的减少。其次，我们引入了一个新的图像分解模型，该模型具有分数拉普拉斯和Riesz电位。我们为唯一解决方案提供了明确的公式，数值实验说明了效率。
In this paper, we focus on learning optimal parameters for PDE-based image regularization and decomposition. 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 other machine learning approaches. The numerical experiments correlate with our theoretical model setting and show a reduction of computing time in contrast to the ROF model. Second we introduce a new 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.
