Minimizing over norms on the gradient.

In this paper, we study the minimization on the gradient for imaging applications. Several recent works have demonstrated that is better than the norm when approximating the norm to promote sparsity. Consequently, we postulate that applying on the gradient is better than the classic total variation (the norm on the gradient) to enforce the sparsity of the image gradient. Numerically, we design a specific splitting scheme, under which we can prove subsequential and global convergence for the alternating direction method of multipliers (ADMM) under certain conditions. Experimentally, we demonstrate visible improvements of over and other nonconvex regularizations for image recovery from low-frequency measurements and two medical applications of MRI and CT reconstruction. Finally, we reveal some empirical evidence on the superiority of over when recovering piecewise constant signals from low-frequency measurements to shed light on future works.