Minimizing L1 over L2 norms on the gradient

TL;DR

This paper investigates the use of L1/L2 minimization on image gradients, demonstrating its superiority over traditional L1 regularization in promoting sparsity and improving image recovery in low-frequency and medical imaging applications.

Contribution

It introduces a novel L1/L2 gradient regularization method, provides a convergence proof for the ADMM algorithm, and empirically shows improved image reconstruction results.

Findings

L1/L2 regularization outperforms L1 in recovering piecewise constant signals.

The proposed method improves MRI and CT image reconstruction quality.

Convergence of the ADMM algorithm is established under certain conditions.

Abstract

In this paper, we study the L1/L2 minimization on the gradient for imaging applications. Several recent works have demonstrated that L1/L2 is better than the L1 norm when approximating the L0 norm to promote sparsity. Consequently, we postulate that applying L1/L2 on the gradient is better than the classic total variation (the L1 norm on the gradient) to enforce the sparsity of the image gradient. To verify our hypothesis, we consider a constrained formulation to reveal empirical evidence on the superiority of L1/L2 over L1 when recovering piecewise constant signals from low-frequency measurements. 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 L1/L2 over L1 and other nonconvex regularizations for image recovery from low-frequency measurements and two medical applications of MRI and CT reconstruction. All the numerical results show the efficiency of our proposed approach.