Digital image inpainting is an interpolation problem, inferring the content in the missing (unknown) region to agree with the known region data such that the interpolated result fulfills some prior knowledge. Low-rank and nonlocal self-similarity are two important priors for image inpainting. Based on the nonlocal self-similarity assumption, an image is divided into overlapped square target patches (submatrices) and the similar patches of any target patch are reshaped as vectors and stacked into a patch matrix. Such a patch matrix usually enjoys a property of low rank or approximately low rank, and its missing entries are recoveried by low-rank matrix approximation (LRMA) algorithms. Traditionally, $n$ nearest neighbor similar patches are searched within a local window centered at a target patch. However, for an image with missing lines, the generated patch matrix is prone to having entirely-missing rows such that the downstream low-rank model fails to reconstruct it well. To address this problem, we propose a region-wise matching (RwM) algorithm by dividing the neighborhood of a target patch into multiple subregions and then search the most similar one within each subregion. A non-convex weighted low-rank decomposition (NC-WLRD) model for LRMA is also proposed to reconstruct all degraded patch matrices grouped by the proposed RwM algorithm. We solve the proposed NC-WLRD model by the alternating direction method of multipliers (ADMM) and analyze the convergence in detail. Numerous experiments on line inpainting (entire-row/column missing) demonstrate the superiority of our method over other competitive inpainting algorithms. Unlike other low-rank-based matrix completion methods and inpainting algorithms, the proposed model NC-WLRD is also effective for removing random-valued impulse noise and structural noise (stripes).

翻译：数字图像修复是一种插值问题，通过推断缺失（未知）区域的内容使其与已知区域数据一致，从而使插值结果满足某些先验知识。低秩性和非局部自相似性是图像修复的两个重要先验。基于非局部自相似性假设，图像被分割为重叠的方形目标块（子矩阵），并将任意目标块的相似块重塑为向量并堆叠成块矩阵。此类块矩阵通常具有低秩或近似低秩特性，其缺失项可通过低秩矩阵近似算法恢复。传统方法在以目标块为中心的局部窗口内搜索$n$个最近邻相似块。然而，对于存在行缺失的图像，生成的块矩阵容易产生整行缺失的问题，导致下游低秩模型难以有效重建。为解决此问题，我们提出一种区域匹配算法，将目标块的邻域划分为多个子区域，并在每个子区域内搜索最相似块。同时，针对低秩矩阵近似任务，提出非凸加权低秩分解模型，用于重建由区域匹配算法分组的所有退化块矩阵。通过交替方向乘数法求解所提出的非凸加权低秩分解模型，并详细分析其收敛性。大量针对行缺失（整行/整列缺失）的修复实验表明，该方法优于其他竞争性修复算法。与基于低秩的其他矩阵补全方法和修复算法不同，所提非凸加权低秩分解模型对随机值脉冲噪声和结构噪声（条纹噪声）同样具有有效去除能力。