# Minimizing L1 over L2 norms on the gradient

arXiv: https://arxiv.org/abs/2101.00809 · HTML: https://mlsanity.com/paper/2101.00809/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 60 references that are in the map.
- First posted on arXiv 4 Jan 2021.

## Builds on (most cited first, 7 of 7)

- [Global Convergence of ADMM in Nonconvex Nonsmooth Optimization](https://mlsanity.com/paper/1511.06324/) (2015)
- [Global convergence of splitting methods for nonconvex composite optimization](https://mlsanity.com/paper/1407.0753/) (2014)
- [Fast L1-L2 minimization via a proximal operator](https://mlsanity.com/paper/1609.09530/) (2016)
- [A Scale Invariant Approach for Sparse Signal Recovery](https://mlsanity.com/paper/1812.08852/) (2018)
- [Limited-angle CT reconstruction via the L1/L2 minimization](https://mlsanity.com/paper/2006.00601/) (2020)
- [Super-Resolution of Positive Sources: the Discrete Setup](https://mlsanity.com/paper/1504.00717/) (2015)
- [A Total Fractional-Order Variation Model for Image Restoration with Non-homogeneous Boundary Conditions and its Numerical Solution](https://mlsanity.com/paper/1509.04237/) (2015)

## Built on by (most cited first, 3 of 3)

- [An Efficient Smoothing and Thresholding Image Segmentation Framework with Weighted Anisotropic-Isotropic Total Variation](https://mlsanity.com/paper/2202.10115/) (2022)
- [Quaternion Nuclear Norms Over Frobenius Norms Minimization for Robust Matrix Completion](https://mlsanity.com/paper/2504.21468/) (2025)
- [Difference of Anisotropic and Isotropic TV for Segmentation under Blur and Poisson Noise](https://mlsanity.com/paper/2301.03393/) (2023)

## Abstract (arXiv)

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.