Image Denoising Using Transformed L1 (TL1) Regularization via ADMM

TL;DR

The paper addresses image denoising with the limitation of TV's convex $\ell_1$ leading to staircase artifacts and reduced contrast. It introduces a TL1 gradient regularizer $\mathrm{TL1}_a$ and solves the resulting nonconvex optimization via ADMM, featuring a closed-form TL1 proximal operator and an FFT-based $\mathbf{u}$-update under periodic boundaries. The method shows superior edge preservation and texture retention compared to $\ell_1$-TV, $\ell_1$--$\ell_2$, MCP, and LOG+TV on benchmark images, with favorable SSIM/PSNR tradeoffs. The approach offers a flexible, nonconvex alternative for gradient regularization with potential extensions to deblurring, inpainting, and super-resolution, aided by efficient per-iteration updates.

Abstract

Total variation (TV) regularization is a classical tool for image denoising, but its convex $\ell_1$ formulation often leads to staircase artifacts and loss of contrast. To address these issues, we introduce the Transformed $\ell_1$ (TL1) regularizer applied to image gradients. In particular, we develop a TL1-regularized denoising model and solve it using the Alternating Direction Method of Multipliers (ADMM), featuring a closed-form TL1 proximal operator and an FFT-based image update under periodic boundary conditions. Experimental results demonstrate that our approach achieves superior denoising performance, effectively suppressing noise while preserving edges and enhancing image contrast.

Figures (5)

• Figure 1: Clean benchmark images: (left) Shapes, (center) Peppers, and (right) Cameraman.
• Figure 2: Denoising results for Shapes. From left to right: Noisy, MCP, and TL1.
• Figure 3: Denoising results for Peppers. From left to right: Noisy, MCP, and TL1.
• Figure 4: Denoising results for Cameraman.
• Figure 5: Zoomed patches comparing texture and edge preservation. TL1 maintains fine details better than competing methods.