On the local and global minimizers of gradient regularized model with box constraints for image restoration

Abstract

Recently, nonconvex and nonsmooth models such as those using ‘norm’ have drawn much attention in the area of image restoration. This work investigates the local and global minimizers of the gradient regularized model with box constraints. There are four major ingredients. Firstly, we show that the set of local minimizers can be represented by solutions to some quadratic problems, which are independent of the fidelity parameter α. Based on this, every point satisfying the first-order necessary condition is a local minimizer. Secondly, any two local minimizers have different energy values under certain assumptions, implying the uniqueness of the global minimizer. Thirdly, there exists a uniform lower bound for nonzero gradients of the restored images. Finally, we show that the global minimizer set is piecewise constant in terms of α, and when A is of full column rank and α is large enough, the distance between the true image and the restored images is bounded by the noise level. The numerical examples perfectly demonstrate our theoretical analysis.