A new algorithm framework for image inpainting in transform domain

In this paper, we focus on variational approaches for image inpainting in transform domain and propose two new algorithms, iterative coupled transform domain inpainting (ICTDI) and iterative decoupled transform domain inpainting. In the derivation of ICTDI, we use operator splitting and the quadratic penalty technique to get a new approximate problem of the basic model. By the alternating minimization method, the approximate problem can be decomposed as three relatively simple subproblems with closed-form solutions. However, ICTDI is not efficient when some adaptive regularization operator is used, such as the learned BM3D frame. To overcome this drawback, with some modifications, we decouple our framework into three relatively independent parts: denoising, linear combination in the transform domain, and linear combination in the image domain. Therefore, we can use any existing denoising method in the denoising step. We consider three choices for regularization operators in our approach: gradient operator, tight framelet transform, and learned BM3D frame. The numerical experiments and comparisons on various images demonstrate the effectiveness of the proposed methods. The convergence of the numerical algorithms is proved under some assumptions.