论文标题
分数laplacian正则化水平集合的收敛性
Convergence of level sets in fractional Laplacian regularization
论文作者
论文摘要
分数拉普拉斯在图像降解和逆问题的正则化中的使用享有最近的流行，因为对于不连续的功能，它的行为比基于$ h^1 $规范的方法的积极性较小，而使用线性和可计算的，可以使用快速的数值方法进行计算。在这项工作中，我们检查了在消失的噪声和正则化参数方面与分数拉普拉斯定于定期的脱氧和线性反问题。在第一种情况下，将清洁数据假定为分段常数，并且在第二种情况下连续并满足源条件。在这些情况下，我们证明了相对于Hausdorff距离的水平设置边界收敛的结果，以及在Denoising和Indifatrix清洁数据的情况下，其他收敛速率的结果。用于此目的的主要技术工具是由Savin和Valdinoci建立的一系列障碍，用于研究分数Allen-Cahn方程。为了帮助将这些分数方法置于上下文中，整个过程中都提供了与总变化和经典的拉普拉斯人的比较。
The use of the fractional Laplacian in image denoising and regularization of inverse problems has enjoyed a recent surge in popularity, since for discontinuous functions it can behave less aggressively than methods based on $H^1$ norms, while being linear and computable with fast spectral numerical methods. In this work, we examine denoising and linear inverse problems regularized with fractional Laplacian in the vanishing noise and regularization parameter regime. The clean data is assumed piecewise constant in the first case, and continuous and satisfying a source condition in the second. In these settings, we prove results of convergence of level set boundaries with respect to Hausdorff distance, and additionally convergence rates in the case of denoising and indicatrix clean data. The main technical tool for this purpose is a family of barriers constructed by Savin and Valdinoci for studying the fractional Allen-Cahn equation. To help put these fractional methods in context, comparisons with the total variation and classical Laplacian are provided throughout.
