To keep structures in the restoration problem is very important via coupling the local information of the image with the proposed model. In this paper we propose a local self-adaptive ${\ell }^p$-regularization model for $p\in (0,2)$ based on the total variation scheme, where the choice of $p$ depends on the local structures described by the eigenvalues of the structure tensor. Since the proposed model as the classic ${\ell }^p$ problem unifies two classes of optimization problems such as the nonconvex and nonsmooth problem when $p\in (0,1)$, and the convex and smooth problem when $p\in (1,2)$, it is generally challenging to find a ready algorithmic framework to solve it. Here we propose a new and robust numerical method via coupling with the half-quadratic scheme and the alternating direction method of multipliers (ADMM). The convergence of the proposed algorithm is established and the numerical experiments illustrate the possible advantages of the proposed model and numerical methods over some existing variational-based models and methods.

通过将图像的局部信息与所提出的模型耦合，使结构保持在修复问题中非常重要。在本文中，我们提出了一种局部自适应${\ell }^p$的正则化模型 $p\in （0，2）$ 基于总变化方案，其中选择 $p$取决于由结构张量的特征值描述的局部结构。自从提出模型作为经典${\ell }^p$ 问题统一了两类优化问题，例如非凸问题和非光滑问题 $p\in （0，\mathrm{1个}）$，以及凸和光滑问题 $p\in （\mathrm{1个}，2）$，通常很难找到一个好的算法框架来解决它。在这里，我们通过与半二次方案和乘数交替方向方法（ADMM）耦合，提出了一种新的且鲁棒的数值方法。建立了所提算法的收敛性，数值实验证明了所提模型和数值方法相对于一些现有的基于变分的模型和方法的可能优势。