# Transformed ℓ1 Gradient Regularization for Image Denoising

arXiv: https://arxiv.org/abs/2511.15060 · HTML: https://mlsanity.com/paper/2511.15060/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 40 references that are in the map.
- First posted on arXiv 19 Nov 2025.

## Builds on (most cited first, 8 of 8)

- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [Fast L1-L2 minimization via a proximal operator](https://mlsanity.com/paper/1609.09530/) (2016)
- [The proximal alternating direction method of multipliers in the nonconvex setting: convergence analysis and rates](https://mlsanity.com/paper/1801.01994/) (2018)
- [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)
- [A Novel Regularization Based on the Error Function for Sparse Recovery](https://mlsanity.com/paper/2007.02784/) (2020)
- [Transformed Schatten-1 Iterative Thresholding Algorithms for Low Rank Matrix Completion](https://mlsanity.com/paper/1506.04444/) (2015)
- [Signal and Image Recovery with Scale and Signed Permutation Invariant Sparsity-Promoting Functions](https://mlsanity.com/paper/2511.05777/) (2025)

## Abstract (arXiv)

Total variation (TV) regularization is a classical edge-preserving technique widely used across image recovery and reconstruction problems; however, its convex ℓ1 gradient penalty tends to over-shrink large gradients, producing staircase artifacts and contrast loss. We propose a gradient-based regularization using the Transformed ℓ1 (TL1) penalty and apply it to image denoising. The TL1 penalty asymptotically interpolates between ℓ1 and the ℓ0 pseudo-norm, offering a principled alternative to TV that better preserves sharp edges and piecewise-smooth regions. Moreover, TL1 admits a tractable proximal operator, enabling an efficient algorithm based on a proximal splitting scheme with subproblems solved by the Alternating Direction Method of Multipliers (ADMM). The weak convexity of TL1 guarantees global convergence of the proximal iterates to a stationary point under mild conditions. Numerical experiments on image denoising demonstrate that the proposed method effectively preserves sharp edges, local contrast, and piecewise-smooth structures, outperforming other gradient-based approaches.