ERD: Exponential Retinex decomposition based on weak space and hybrid nonconvex regularization and its denoising application

TL;DR

This work tackles denoising under additive noise by recasting Retinex-based decomposition into an exponential Retinex framework that separates oscillation, illumination, and reflection components. It introduces a weak-space $H^{-1}$ fidelity term for the oscillation and two hybrid nonconvex regularizers on the illumination and reflection components, solved via an ADMM solver augmented with Majorize-Minimization. The authors prove convergence under standard assumptions and demonstrate superior PSNR and MSSIM across gray, color, and medical images, as well as multiple datasets, highlighting robust edge and texture preservation. The proposed approach offers a principled, three-component decomposition for denoising with strong empirical performance and potential applicability to a broad range of imaging problems.

Abstract

The Retinex theory models the image as a product of illumination and reflection components, which has received extensive attention and is widely used in image enhancement, segmentation and color restoration. However, it has been rarely used in additive noise removal due to the inclusion of both multiplication and addition operations in the Retinex noisy image modeling. In this paper, we propose an exponential Retinex decomposition model based on hybrid non-convex regularization and weak space oscillation-modeling for image denoising. The proposed model utilizes non-convex first-order total variation (TV) and non-convex second-order TV to regularize the reflection component and the illumination component, respectively, and employs weak $H^{-1}$ norm to measure the residual component. By utilizing different regularizers, the proposed model effectively decomposes the image into reflection, illumination, and noise components. An alternating direction multipliers method (ADMM) combined with the Majorize-Minimization (MM) algorithm is developed to solve the proposed model. Furthermore, we provide a detailed proof of the convergence property of the algorithm. Numerical experiments validate both the proposed model and algorithm. Compared with several state-of-the-art denoising models, the proposed model exhibits superior performance in terms of peak signal-to-noise ratio (PSNR) and mean structural similarity (MSSIM).

Theorems & Definitions (17)

• Lemma 2.1
• Theorem 3.1
• Proof
• Theorem 3.5
• Proof
• Theorem 3.6
• Proof
• Theorem 3.7
• Proof
• Theorem 3.8