Minimizing L ₁ over L ₂ norms on the gradient

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 MRI and CT 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.