A Nonconvex Nonsmooth Image Prior Based on the Hyperbolic Tangent Function

In this paper, we propose a nonconvex and nonsmooth image prior based on the hyperbolic tangent function and apply it as a regularization term for image restoration and reconstruction problems. Theoretically, we analyze the properties of the function and the minimizers of its associated proximal problem. Since the proximal problem has no closed-form solution, we propose a derivative-free Nelder-Mead simplex based selection algorithm to find the global minimizer. To reduce the computational cost, we only solve a small 1D problem and then use the 1D solution template as a look-up table to interpolate high-dimension data. Moreover, we consider a variational model based on the proposed image prior. Then we use the alternating direction method of multipliers algorithm on the nonconvex model to derive efficient numerical algorithms. Various experiments on image denoising, image deblurring, and image reconstruction demonstrate that the proposed nonconvex prior is competitive with the existing priors. In particular, it outperforms others in recovering piecewise constant images.