Transformed $\ell_1$ Gradient Regularization for Image Denoising

Abstract: Total variation (TV) regularization is a classical edge-preserving technique widely used across image recovery and reconstruction problems; however, its convex $\ell_1$ gradient penalty tends to over-shrink large gradients, producing staircase artifacts and contrast loss. We propose a gradient-based regularization using the Transformed $\ell_1$ (TL1) penalty and apply it to image denoising. The TL1 penalty asymptotically interpolates between $\ell_1$ and the $\ell_0$ 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.