Inexact alternating direction method based on proximity projection operator for image inpainting in wavelet domain

Wavelet domain inpainting refers to the process of recovering the missing coefficients during image compression or transmission stage. Recently, efficient algorithms including the augmented Lagrangian methods (ALMs) and the primal-dual hybrid gradient (PDHG) algorithm have been successfully applied to solve the classical variational model of wavelet inpainting. However, the standard ALMs and PDHG require the wavelet bases to be orthogonal. Very recently, an approximated PDHG algorithm was proposed to deal with the case of the non-orthogonal wavelet such as Daubechies 7–9 wavelet. In this paper, we further develop an inexact alternating direction method for solving the general non-orthogonal wavelet inpainting problem, where one sub-minimization problem is solved by the proximity projection algorithm based on the Newton iteration. The global convergence of the proposed algorithm is further investigated under mild conditions. Numerical experiments demonstrate that the proposed method overall outperforms the recently proposed approximated PDHG algorithm, especially in the computational time.