An inexact proximal majorization–minimization method for a class of image reconstruction models

Abstract We propose a class of nonconvex and nonsmooth image reconstruction models for impulse noise removal, which is rather general and covers existing nonconvex and nonsmooth models for reconstructing images with impulse noise. For this class of models, we develop an inexact proximal majorization–minimization algorithm with an implementable inexactness criterion by constructing a strongly convex local majorization of the objective function and meanwhile seeking its inexact minimizer in each iteration, and provide its theoretical certificate by proving the full convergence of the iterate sequence under the Kurdyka–Łojasiewicz property of the constructed potential function. This inexact proximal majorization–minimization method is applied to linear image restoration problems from deblurring and inpainting, and a class of nonlinear image reconstruction from Fourier phase retrieval (PR). Numerical comparisons with the proximal linearized minimization algorithm for image deblurring and the alternating direction method of multipliers for image inpainting and Fourier PR validate the efficiency of the proposed method.