Simultaneously inpainting in image and transformed domains

In this paper, we focus on the restoration of images that have incomplete data in either the image domain or the transformed domain or in both. The transform used can be any orthonormal or tight frame transforms such as orthonormal wavelets, tight framelets, the discrete Fourier transform, the Gabor transform, the discrete cosine transform, and the discrete local cosine transform. We propose an iterative algorithm that can restore the incomplete data in both domains simultaneously. We prove the convergence of the algorithm and derive the optimal properties of its limit. The algorithm generalizes, unifies, and simplifies the inpainting algorithm in image domains given in Cai et al. (Appl Comput Harmon Anal 24:131-149, 2008) and the inpainting algorithms in the transformed domains given in Cai et al. (SIAM J Sci Comput 30(3):1205-1227, 2008), Chan et al. (SIAM J Sci Comput 24:1408-1432, 2003; Appl Comput Harmon Anal 17:91-115, 2004). Finally, applications of the new algorithm to super-resolution image reconstruction with different zooms are presented.