Minimizing L 1 over L 2 norms on the gradient

Abstract

Abstract In this paper, we study the L 1/L 2 minimization on the gradient for imaging applications. Several recent works have demonstrated that L 1/L 2 is better than the L 1 norm when approximating the L 0 norm to promote sparsity. Consequently, we postulate that applying L 1/L 2 on the gradient is better than the classic total variation (the L 1 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 L 1/L 2 over L 1 and other nonconvex regularizations for image recovery from low-frequency measurements and two medical applications of magnetic resonance imaging and computed tomography reconstruction. Finally, we reveal some empirical evidence on the superiority of L 1/L 2 over L 1 when recovering piecewise constant signals from low-frequency measurements to shed light on future works.