# Plug-and-Play Image Restoration with Flow Matching: A Continuous Viewpoint

arXiv: https://arxiv.org/abs/2512.04283 · HTML: https://mlsanity.com/paper/2512.04283/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Inverse problems and image restoration, one of 14 directions in Diffusion models.
- Builds on 36 of its 52 references that are in the map.
- First posted on arXiv 3 Dec 2025.

## Placed in

- Diffusion models / [Inverse problems and image restoration](https://mlsanity.com/diffusion/inverse-problems-and-image-restoration/)

## Builds on (most cited first, 30 of 36)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Denoising Diffusion Models for Plug-and-Play Image Restoration](https://mlsanity.com/paper/2305.08995/) (2023)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Come-Closer-Diffuse-Faster: Accelerating Conditional Diffusion Models for Inverse Problems through Stochastic Contraction](https://mlsanity.com/paper/2112.05146/) (2021)
- [Visual Prompting via Image Inpainting](https://mlsanity.com/paper/2209.00647/) (2022)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [Plug-and-Play Methods Provably Converge with Properly Trained Denoisers](https://mlsanity.com/paper/1905.05406/) (2019)
- [Deblurring by Realistic Blurring](https://mlsanity.com/paper/2004.01860/) (2020)
- [Semantic Image Inpainting with Deep Generative Models](https://mlsanity.com/paper/1607.07539/) (2016)
- [The probability flow ODE is provably fast](https://mlsanity.com/paper/2305.11798/) (2023)
- [Unfolding the Alternating Optimization for Blind Super Resolution](https://mlsanity.com/paper/2010.02631/) (2020)
- [D-Flow: Differentiating through Flows for Controlled Generation](https://mlsanity.com/paper/2402.14017/) (2024)
- [Gradient Step Denoiser for convergent Plug-and-Play](https://mlsanity.com/paper/2110.03220/) (2021)
- [Principled Probabilistic Imaging using Diffusion Models as Plug-and-Play Priors](https://mlsanity.com/paper/2405.18782/) (2024)
- [PnP-Flow: Plug-and-Play Image Restoration with Flow Matching](https://mlsanity.com/paper/2410.02423/) (2024)
- [Deep Image Deblurring: A Survey](https://mlsanity.com/paper/2201.10700/) (2022)
- [Maximum Likelihood Training for Score-Based Diffusion ODEs by High-Order Denoising Score Matching](https://mlsanity.com/paper/2206.08265/) (2022)
- [Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstruction](https://mlsanity.com/paper/2403.17042/) (2024)
- [Training-free Linear Image Inverses via Flows](https://mlsanity.com/paper/2310.04432/) (2023)
- [Image Denoising: The Deep Learning Revolution and Beyond -- A Survey Paper --](https://mlsanity.com/paper/2301.03362/) (2023)
- [Regularization by Denoising via Fixed-Point Projection (RED-PRO)](https://mlsanity.com/paper/2008.00226/) (2020)
- [Flow Priors for Linear Inverse Problems via Iterative Corrupted Trajectory Matching](https://mlsanity.com/paper/2405.18816/) (2024)
- [NAN: Noise-Aware NeRFs for Burst-Denoising](https://mlsanity.com/paper/2204.04668/) (2022)

## Built on by (most cited first, 3 of 3)

- [Second Order Drifting Models](https://mlsanity.com/paper/2608.07924/) (2026)
- [Deep learning methods for inverse problems using connections between proximal operators and Hamilton-Jacobi equations](https://mlsanity.com/paper/2512.23829/) (2025)
- [SurfSpec: Enhancing Off-Target-Agnostic Specificity by Bounding Pocket-Ligand Geometric Mismatch](https://mlsanity.com/paper/2609.02963/) (2026)

## Abstract (arXiv)

Flow matching-based generative models have been integrated into the plug-and-play image restoration framework, and the resulting plug-and-play flow matching (PnP-Flow) model has achieved some remarkable empirical success for image restoration. However, the theoretical understanding of PnP-Flow lags its empirical success. In this paper, we derive a continuous limit for PnP-Flow, resulting in a stochastic differential equation (SDE) surrogate model of PnP-Flow. The SDE model provides two particular insights to improve PnP-Flow for image restoration: (1) It enables us to quantify the error for image restoration, informing us to improve step scheduling and regularize the Lipschitz constant of the neural network-parameterized vector field for error reduction. (2) It informs us to accelerate off-the-shelf PnP-Flow models via extrapolation, resulting in a rescaled version of the proposed SDE model. We validate the efficacy of the SDE-informed improved PnP-Flow using several benchmark tasks, including image denoising, deblurring, super-resolution, and inpainting. Numerical results show that our method significantly outperforms the baseline PnP-Flow and other state-of-the-art approaches, achieving superior performance across evaluation metrics.