# Transformed $\ell_1$ Gradient Regularization for Image Denoising

**Authors:** Nabiha Choudhury, Jianqing Jia, Yifei Lou  
**arXiv:** [2511.15060](https://arxiv.org/abs/2511.15060)  
**Date:** 2025-11-19  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2511.15060v2/

## Key points

- Total variation (TV) regularization is popular in image recovery for preserving edges
- Traditional convex $\ell_1$ gradient penalty in TV regularization can lead to over-shrinkage of large gradients, causing undesirable artifacts and loss of contrast
- Transformed $\ell_1$ (TL1) penalty proposed by Nabiha Choudhury, Jianqing Jia, and Yifei Lou offers an alternative that interpolates between $\ell_1$ norm and $\ell_0$ pseudo-norm
- TL1 better preserves sharp edges and piecewise-smooth regions in images compared to traditional TV regularization
- TL1 has a tractable proximal operator, enabling efficient algorithm development using ADMM
- TL1's weak convexity property ensures global convergence of proximal iterates towards a stationary point under mild conditions, enhancing denoising stability and reliability
- Numerical experiments show TL1-based method excels at preserving sharp edges, enhancing local contrast, and capturing piecewise-smooth structures within images
- TL1 regularization method is superior to other gradient-based techniques for achieving high-quality denoised images with minimal artifacts

## Summary

Total variation (TV) regularization has long been a popular technique in image recovery and reconstruction due to its ability to preserve edges. However, the traditional convex $\ell_1$ gradient penalty used in TV regularization often leads to the over-shrinkage of large gradients, resulting in undesirable staircase artifacts and loss of contrast in images. In response to this limitation, a novel approach utilizing Transformed $\ell_1$ (TL1) penalty for gradient-based regularization has been proposed by Nabiha Choudhury, Jianqing Jia, and Yifei Lou. The TL1 penalty offers a unique advantage by asymptotically interpolating between the $\ell_1$ norm and the $\ell_0$ pseudo-norm. This interpolation provides a principled alternative to TV regularization that better preserves sharp edges and piecewise-smooth regions in images. Additionally, TL1 comes with a tractable proximal operator, facilitating the development of an efficient algorithm based on a proximal splitting scheme with subproblems solved using the Alternating Direction Method of Multipliers (ADMM). One key feature of TL1 is its weak convexity property, ensuring global convergence of the proximal iterates towards a stationary point under mild conditions. This property enhances the stability and reliability of the denoising process when using TL1 regularization. Numerical experiments conducted on image denoising tasks have demonstrated the effectiveness of the proposed TL1-based method. The results show that this approach excels at preserving sharp edges, enhancing local contrast, and accurately capturing piecewise-smooth structures within images. In comparison to other gradient-based techniques, the TL1 regularization method stands out as a superior choice for achieving high-quality denoised images with minimal artifacts. In conclusion,Nabiha Choudhury, Jianqing Jia, and Yifei Lou's work on Transformed $\ell_1$ Gradient Regularization represents a significant advancement in image denoising techniques. By addressing the limitations of traditional TV regularization through innovative use of TL1 penalty, their method offers improved performance in preserving image features while effectively reducing noise levels.