Convex MR brain image reconstruction via non‐convex total variation minimization

Total variation (TV) regularization is a technique commonly utilized to promote sparsity of image in gradient domain. In this article, we address the problem of MR brain image reconstruction from highly undersampled Fourier measurements. We define the Moreau enhanced function of L1 norm, and introduce the minmax‐concave TV (MCTV) penalty as a regularization term for MR brain image reconstruction. MCTV strongly induces the sparsity in gradient domain, and fits the frame of fast algorithms (eg, ADMM) for solving optimization problems. Although MCTV is non‐convex, the cost function in each iteration step can maintain convexity by specifying the relative nonconvexity parameter properly. Experimental results demonstrate the superior performance of the proposed method in comparison with standard TV as well as non‐local TV minimization method, which suggests that MCTV may have promising applications in the field of neuroscience in the future.