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内插问题,它通过寻找与已知区域数据一致的缺失(未知)区域来推断内容,从而使内插结果符合某些先验知识。低秩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的假设,对图像进行重叠的正方形目标块(子矩阵)划分,然后将任何目标块的相似块重塑为向量并叠加在一个块矩阵中。这样的块矩阵通常具有低秩性质或近似低秩性质,并且其缺失条目通过低秩矩阵逼近(LRMA)算法进行恢复。传统上,在一个局部窗口内寻找$n$个最近邻相似块。然而,对于缺失行的图像,生成的块矩阵容易具有整行缺失的问题,从而导致下游的低秩模型无法很好地重构它。为了解决这个问题,我们提出了一种区域匹配(RwM)算法,通过将目标块的邻域划分为多个子区域,然后在每个子区域中寻找最相似的块。我们还提出了一种非凸加权低秩分解(NC-WLRD)模型,用于重构由提出的RwM算法分组的所有降级块矩阵。我们通过交替方向乘子(ADMM)求解提出的NC-WLRD模型,并详细分析了其收敛性。大量的行修复(整行/列缺失)实验证明了我们的方法优于其他竞争性的修复算法。与其他基于低秩矩阵完成方法和修复算法不同,所提出的模型NC-WLRD也适用于去除随机脉冲噪声和结构噪声(条纹)。