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

• Type of Material: Article
• 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
• Language: English
• Date: 2022-06-01
• Publisher: IOP Publishing Ltd
• Publication Version: Author Accepted Manuscript (After Peer Review)
• Copyright Statement: © 2022 IOP Publishing Ltd
• License: Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International
• Final Published Version (URL): http://dx.doi.org/10.1088/1361-6420/ac64fb
• Title of Journal or Parent Work: INVERSE PROBLEMS
• Volume: 38
• Issue: 6
• Keywords: ANGLE CT RECONSTRUCTION; SUPERRESOLUTION; REPRESENTATION; Mathematics; Mathematics, Applied; global convergence; Physics, Mathematical; L (2) minimization; alternating direction method of multipliers; Physical Sciences; PENALIZED LIKELIHOOD; Physics; Science & Technology; piecewise constant images; VARIABLE SELECTION; RECOVERY; PENALTY; L (1); MODEL; MINIMIZATION
• Research Categories: Mathematics; Statistics

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.