Total variation (TV) denoising has attracted considerable attention in 1-D and 2-D signal processing. For image denoising, the convex cost function can be viewed as the regularized linear least squares problem (1 regularizer for anisotropic case and 2 regularizer for isotropic case). However, these convex regularizers often underestimate the high-amplitude components of the true image. In this paper, non-convex regularizers for 2-D TV denoising models are proposed. These regularizers are based on the Moreau envelope and minimax-concave penalty, which can maintain the convexity of the cost functions. Then, efficient algorithms based on forward-backward splitting are proposed to solve the new cost functions. The numerical results show the effectiveness of the proposed non-convex regularizers for both synthetic and real-world image. INDEX TERMS Total variation denoising, 1 norm, 2 norm, non-convex regularizer, forward-backward splitting.

Magnetic resonance imaging (MRI) reconstruction model based on total variation (TV) regularization can deal with problems such as incomplete reconstruction, blurred boundary, and residual noise. In this article, a non‐convex isotropic TV regularization reconstruction model is proposed to overcome the drawback. Moreau envelope and minmax‐concave penalty are firstly used to construct the non‐convex regularization of L2 norm, then it is applied into the TV regularization to construct the sparse reconstruction model. The proposed model can extract the edge contour of the target effectively since it can avoid the underestimation of larger nonzero elements in convex regularization. In addition, the global convexity of the cost function can be guaranteed under certain conditions. Then, an efficient algorithm such as alternating direction method of multipliers is proposed to solve the new cost function. Experimental results show that, compared with several typical image reconstruction methods, the proposed model performs better. Both the relative error and the peak signal‐to‐noise ratio are significantly improved, and the reconstructed images also show better visual effects. The competitive experimental results indicate that the proposed approach is not limited to MRI reconstruction, but it is general enough to be used in other fields with natural images.