Minimizing L (1) over L (2) norms on the gradient

• Authors: Chao Wang, Southern University of Science & Technology; Min Tao, Nanjing University; Chen-Nee Chuah, University of California Davis; James Nagy, Emory University; Yifei Lou, University of Texas Dallas
• Journal Title: INVERSE PROBLEMS
• Volume: Volume 38, Number 6
• Publisher: IOP Publishing Ltd | 2022-06-01
• Type of Work: Article | Post-print: After Peer Review
• Publisher DOI: http://doi.org/10.1088/1361-6420/ac64fb
• Subjects: Mathematics; Engineering, Electronics and Electrical; Statistics
• Keywords: Science & Technology; Physical Sciences; Mathematics, Applied; Physics, Mathematical; Mathematics; Physics; L (1); L (2) minimization; piecewise constant images; alternating direction method of multipliers; global convergence; ANGLE CT RECONSTRUCTION; PENALIZED LIKELIHOOD; VARIABLE SELECTION; MODEL; SUPERRESOLUTION; REPRESENTATION; MINIMIZATION; RECOVERY; PENALTY

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.