Image Restoration Models with Optimal Transport and Total Variation Regularization

TL;DR

This work introduces image restoration models that fuse total variation regularization with a Wasserstein-1–based data fidelity, leveraging the dual Lipschitz norm and connections to Meyer's G-norm for cartoon-texture decomposition. The authors develop a practical algorithm combining Primal-Dual Hybrid Gradient (PDHG) updates for the OT component with an augmented Lagrangian approach for TV, and they prove existence, uniqueness, and convergence of minimizers. The framework is extended with a nonconvex KR-Log-TV variant and a Meyer-inspired MTV regularization to better preserve contrast and suppress staircasing. Numerical experiments on diverse images illustrate enhanced texture preservation and image contrast over classical ROF and other decomposition approaches, with solid convergence behavior and flexibility in discretization and convolutional extensions.

Abstract

In this paper, we propose image restoration models using optimal transport (OT) and total variation regularization. We present theoretical results of the proposed models based on the relations between the dual Lipschitz norm from OT and the G-norm introduced by Yves Meyer. We design a numerical method based on the Primal-Dual Hybrid Gradient (PDHG) algorithm for the Wasserstain distance and the augmented Lagrangian method (ALM) for the total variation, and the convergence analysis of the proposed numerical method is established. We also consider replacing the total variation in our model by one of its modifications developed in \cite{zhu}, with the aim of suppressing the stair-casing effect and preserving image contrasts. Numerical experiments demonstrate the features of the proposed models.

Figures (7)

• Figure 1: The cartoon and texture decomposition $f=u+v$ obtained from the ROF, $G$-TV \ref{['g-pro-l2']}, and our model \ref{['kr_new_rel']}.
• Figure 2: Results using the ROF model, our model \ref{['kr_new_rel']}, MTV ($a=5/255$) \ref{['log-tv']}, and the proposed model with MTV \ref{['kr_new_rel_log']}. The additive noise $n$ has the norm $\|n\| = 25$. For the proposed models \ref{['kr_new_rel']} and \ref{['kr_new_rel_log']}, we choose parameter $\alpha=100$, and the other parameters are tuned so that the removed noise parts have the same norm $\|f-u\| \approx 25$ as the ROF model.
• Figure 3: Results using the ROF model, our model \ref{['kr_new_rel']}, MTV model ($a=8/255$)\ref{['log-tv']} and the proposed model with MTV \ref{['kr_new_rel_log']}. The additive noise $n$ has norm $\|n\| = 25$. For the proposed models, we choose the same parameter $\alpha=80$ and the other parameters are tuned such that the removed noise part has the norm $\|f-u\| \approx 25$ as the ROF model and MTV model.
• Figure 4: Results using the ROF model, our model \ref{['kr_new_rel']}, MTV ($a=8/255$) \ref{['log-tv']} and the proposed model with MTV \ref{['kr_new_rel_log']}. The additive noise $n$ has the norm $\|n\| = 20$ and the parameters are tuned such that the removed noise part has the norm $\|f-u\| \approx 20$ for all the models.
• Figure 5: Results using the ROF model, and our model \ref{['kr_new_rel']}. The noisy and blurred image is obtained by applying a Gaussian blur $(G,1.6,25)$ and adding a Gaussian noise with $\sigma=15$. The additive noise $n$ has the norm $\|n\|_{L^2} \approx 15$. The parameters are adjusted such that the removed noise parts of all the models have the norm $\|f-K*u\|_{L^2} \approx 15$.

Theorems & Definitions (8)

• lemma thmcounterlemma
• theorem 1
• proof
• lemma thmcounterlemma
• theorem 2
• proof
• proposition thmcounterproposition
• proof