# Image Restoration with Mean-Reverting Stochastic Differential Equations

arXiv: https://arxiv.org/abs/2301.11699 · HTML: https://mlsanity.com/paper/2301.11699/

## Key facts

- Cited by 183 papers in the map (data through 14 Oct 2026).
- Builds on 50 of its 76 references that are in the map.
- First posted on arXiv 27 Jan 2023.

## Builds on (most cited first, 30 of 50)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [SwinIR: Image Restoration Using Swin Transformer](https://mlsanity.com/paper/2108.10257/) (2021)
- [Restormer: Efficient Transformer for High-Resolution Image Restoration](https://mlsanity.com/paper/2111.09881/) (2021)
- [Enhanced Deep Residual Networks for Single Image Super-Resolution](https://mlsanity.com/paper/1707.02921/) (2017)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [RePaint: Inpainting using Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.09865/) (2022)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [Palette: Image-to-Image Diffusion Models](https://mlsanity.com/paper/2111.05826/) (2021)
- [Image Super-Resolution Using Deep Convolutional Networks](https://mlsanity.com/paper/1501.00092/) (2014)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Uformer: A General U-Shaped Transformer for Image Restoration](https://mlsanity.com/paper/2106.03106/) (2021)
- [Multi-Stage Progressive Image Restoration](https://mlsanity.com/paper/2102.02808/) (2021)
- [Deep Multi-scale Convolutional Neural Network for Dynamic Scene Deblurring](https://mlsanity.com/paper/1612.02177/) (2016)
- [Simple Baselines for Image Restoration](https://mlsanity.com/paper/2204.04676/) (2022)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [FFDNet: Toward a Fast and Flexible Solution for CNN based Image Denoising](https://mlsanity.com/paper/1710.04026/) (2017)

## Built on by (most cited first, 30 of 183)

- [ResShift: Efficient Diffusion Model for Image Super-resolution by Residual Shifting](https://mlsanity.com/paper/2307.12348/) (2023)
- [Diffusion Model-Based Image Editing: A Survey](https://mlsanity.com/paper/2402.17525/) (2024)
- [A Survey on Diffusion Models for Inverse Problems](https://mlsanity.com/paper/2410.00083/) (2024)
- [Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restoration](https://mlsanity.com/paper/2303.11435/) (2023)
- [Low-Light Image Enhancement with Wavelet-based Diffusion Models](https://mlsanity.com/paper/2306.00306/) (2023)
- [Controlling Vision-Language Models for Multi-Task Image Restoration](https://mlsanity.com/paper/2310.01018/) (2023)
- [Diffusion Models for Image Restoration and Enhancement: A Comprehensive Survey](https://mlsanity.com/paper/2308.09388/) (2023)
- [Refusion: Enabling Large-Size Realistic Image Restoration with Latent-Space Diffusion Models](https://mlsanity.com/paper/2304.08291/) (2023)
- [Diffusion Models in Low-Level Vision: A Survey](https://mlsanity.com/paper/2406.11138/) (2024)
- [Selective Hourglass Mapping for Universal Image Restoration Based on Diffusion Model](https://mlsanity.com/paper/2403.11157/) (2024)
- [LightenDiffusion: Unsupervised Low-Light Image Enhancement with Latent-Retinex Diffusion Models](https://mlsanity.com/paper/2407.08939/) (2024)
- [Residual Denoising Diffusion Models](https://mlsanity.com/paper/2308.13712/) (2023)
- [Direct Diffusion Bridge using Data Consistency for Inverse Problems](https://mlsanity.com/paper/2305.19809/) (2023)
- [Diffusion Models, Image Super-Resolution And Everything: A Survey](https://mlsanity.com/paper/2401.00736/) (2024)
- [Efficient Diffusion Model for Image Restoration by Residual Shifting](https://mlsanity.com/paper/2403.07319/) (2024)
- [DMPlug: A Plug-in Method for Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2405.16749/) (2024)
- [EDiffSR: An Efficient Diffusion Probabilistic Model for Remote Sensing Image Super-Resolution](https://mlsanity.com/paper/2310.19288/) (2023)
- [Adversarial Diffusion Compression for Real-World Image Super-Resolution](https://mlsanity.com/paper/2411.13383/) (2024)
- [Image Restoration Through Generalized Ornstein-Uhlenbeck Bridge](https://mlsanity.com/paper/2312.10299/) (2023)
- [Reference-guided Controllable Inpainting of Neural Radiance Fields](https://mlsanity.com/paper/2304.09677/) (2023)
- [Structure Matters: Tackling the Semantic Discrepancy in Diffusion Models for Image Inpainting](https://mlsanity.com/paper/2403.19898/) (2024)
- [FoundIR: Unleashing Million-scale Training Data to Advance Foundation Models for Image Restoration](https://mlsanity.com/paper/2412.01427/) (2024)
- [FourierMamba: Fourier Learning Integration with State Space Models for Image Deraining](https://mlsanity.com/paper/2405.19450/) (2024)
- [Entropy-regularized Diffusion Policy with Q-Ensembles for Offline Reinforcement Learning](https://mlsanity.com/paper/2402.04080/) (2024)
- [Conditional Image Synthesis with Diffusion Models: A Survey](https://mlsanity.com/paper/2409.19365/) (2024)
- [NTIRE 2024 Challenge on Low Light Image Enhancement: Methods and Results](https://mlsanity.com/paper/2404.14248/) (2024)
- [Visual-Instructed Degradation Diffusion for All-in-One Image Restoration](https://mlsanity.com/paper/2506.16960/) (2025)
- [Teaching Tailored to Talent: Adverse Weather Restoration via Prompt Pool and Depth-Anything Constraint](https://mlsanity.com/paper/2409.15739/) (2024)
- [Score-based Generative Diffusion Models for Social Recommendations](https://mlsanity.com/paper/2412.15579/) (2024)
- [OFTSR: One-Step Flow for Image Super-Resolution with Tunable Fidelity-Realism Trade-offs](https://mlsanity.com/paper/2412.09465/) (2024)

## Abstract (arXiv)

This paper presents a stochastic differential equation (SDE) approach for general-purpose image restoration. The key construction consists in a mean-reverting SDE that transforms a high-quality image into a degraded counterpart as a mean state with fixed Gaussian noise. Then, by simulating the corresponding reverse-time SDE, we are able to restore the origin of the low-quality image without relying on any task-specific prior knowledge. Crucially, the proposed mean-reverting SDE has a closed-form solution, allowing us to compute the ground truth time-dependent score and learn it with a neural network. Moreover, we propose a maximum likelihood objective to learn an optimal reverse trajectory that stabilizes the training and improves the restoration results. The experiments show that our proposed method achieves highly competitive performance in quantitative comparisons on image deraining, deblurring, and denoising, setting a new state-of-the-art on two deraining datasets. Finally, the general applicability of our approach is further demonstrated via qualitative results on image super-resolution, inpainting, and dehazing. Code is available at https://github.com/Algolzw/image-restoration-sde.