A modified non-convex Cauchy total variation regularization model for image restoration

This study introduces a modified non-convex Cauchy total variation (MCauchyTV) model based on the prior property of image. First, we improve the probability density function of the Cauchy distribution, and propose a general Cauchy penalty function. Then, its proximity operator is derived and has a closed-form solution. Second, a non-convex MCauchyTV model is implemented with the proposed multivariate Cauchy function. Following, the model is solved by an effective alternating direction method of multipliers. Its subproblems can be solved quickly using the fast Fourier transform and the proximity operator. Third, we analyze and prove the conditions under which the MCauchyTV proximity operator preserves convexity, ensuring the exact tuning of the system parameters and the convergence of the ADMM. Finally, the efficiency and viability of the MCauchyTV model are validated by image deblurring and magnetic resonance imaging reconstruction.