Several strategies have been applied for the recovery of the missing parts in an image, with recovery performance depending significantly on the image type and the geometry of missing data. To provide a deeper understanding of such image restoration problem, King et al. recently introduced a rigorous multiscale analysis framework and proved that a shearlet based inpainting approach outperforms methods based on more conventional multiscale representations when missing data are line singularities. In this paper, we extend and improve the analysis of the inpainting problem to the more realistic and more challenging setting of images containing curvilinear singularities. We derive inpainting performance guarantees showing that exact image recovery is achieved if the size of the missing singularity is smaller than the size of the structure elements of appropriate functional representations of the image. Our proof relies critically on the microlocal and sparsity properties of the shearlet representation.

已经应用了几种策略来恢复图像中的缺失部分，恢复性能在很大程度上取决于图像类型和缺失数据的几何形状。为了提供对这种图像恢复问题的更深入的了解，King等人。最近，我们引入了严格的多尺度分析框架，并证明了当缺少数据为线奇点时，基于剪切波的修补方法优于基于更常规多尺度表示的方法。在本文中，我们将修复问题的分析扩展和改进到更实际，更具挑战性的包含曲线奇点的图像设置。我们得出的修复性能保证表明，如果缺失的奇异点的大小小于图像的适当功能表示的结构元素的大小，则可以实现精确的图像恢复。我们的证明主要取决于小波表示的微观局部性和稀疏性。