# Image Restoration Through Generalized Ornstein-Uhlenbeck Bridge

arXiv: https://arxiv.org/abs/2312.10299 · HTML: https://mlsanity.com/paper/2312.10299/

## Key facts

- Cited by 41 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 53 of its 77 references that are in the map.
- First posted on arXiv 16 Dec 2023.

## 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 53)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [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)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [SwinIR: Image Restoration Using Swin Transformer](https://mlsanity.com/paper/2108.10257/) (2021)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Image Super-Resolution Using Deep Convolutional Networks](https://mlsanity.com/paper/1501.00092/) (2014)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Multi-Stage Progressive Image Restoration](https://mlsanity.com/paper/2102.02808/) (2021)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [FFDNet: Toward a Fast and Flexible Solution for CNN based Image Denoising](https://mlsanity.com/paper/1710.04026/) (2017)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [Deep Joint Rain Detection and Removal from a Single Image](https://mlsanity.com/paper/1609.07769/) (2016)
- [MAXIM: Multi-Axis MLP for Image Processing](https://mlsanity.com/paper/2201.02973/) (2022)
- [BBDM: Image-to-image Translation with Brownian Bridge Diffusion Models](https://mlsanity.com/paper/2205.07680/) (2022)
- [I2SB: Image-to-Image Schr"odinger Bridge](https://mlsanity.com/paper/2302.05872/) (2023)
- [Dual Diffusion Implicit Bridges for Image-to-Image Translation](https://mlsanity.com/paper/2203.08382/) (2022)

## Built on by (most cited first, 30 of 41)

- [OFTSR: One-Step Flow for Image Super-Resolution with Tunable Fidelity-Realism Trade-offs](https://mlsanity.com/paper/2412.09465/) (2024)
- [UniDB: A Unified Diffusion Bridge Framework via Stochastic Optimal Control](https://mlsanity.com/paper/2502.05749/) (2025)
- [Photo-Realistic Image Restoration in the Wild with Controlled Vision-Language Models](https://mlsanity.com/paper/2404.09732/) (2024)
- [Residual Diffusion Bridge Model for Image Restoration](https://mlsanity.com/paper/2510.23116/) (2025)
- [Taming Diffusion Models for Image Restoration: A Review](https://mlsanity.com/paper/2409.10353/) (2024)
- [IRBridge: Solving Image Restoration Bridge with Pre-trained Generative Diffusion Models](https://mlsanity.com/paper/2505.24406/) (2025)
- [Pixel super-resolved virtual staining of label-free tissue using diffusion models](https://mlsanity.com/paper/2410.20073/) (2024)
- [HOG-Diff: Higher-Order Guided Diffusion for Graph Generation](https://mlsanity.com/paper/2502.04308/) (2025)
- [DPBridge: Latent Diffusion Bridge for Dense Prediction](https://mlsanity.com/paper/2412.20506/) (2024)
- [Enhanced Control for Diffusion Bridge in Image Restoration](https://mlsanity.com/paper/2408.16303/) (2024)
- [System-Embedded Diffusion Bridge Models](https://mlsanity.com/paper/2506.23726/) (2025)
- [TPGDiff: Hierarchical Triple-Prior Guided Diffusion for Image Restoration](https://mlsanity.com/paper/2601.20306/) (2026)
- [Series-to-Series Diffusion Bridge Model](https://mlsanity.com/paper/2411.04491/) (2024)
- [One-Step Residual Shifting Diffusion for Image Super-Resolution via Distillation](https://mlsanity.com/paper/2503.13358/) (2025)
- [Incomplete Data, Complete Dynamics: A Diffusion Approach](https://mlsanity.com/paper/2509.20098/) (2025)
- [Diffusion Bridge or Flow Matching? A Unifying Framework and Comparative Analysis](https://mlsanity.com/paper/2509.24531/) (2025)
- [Fractional Diffusion Bridge Models](https://mlsanity.com/paper/2511.01795/) (2025)
- [Sample from What You See: Visuomotor Policy Learning via Diffusion Bridge with Observation-Embedded Stochastic Differential Equation](https://mlsanity.com/paper/2512.07212/) (2025)
- [Clear Nights Ahead: Towards Multi-Weather Nighttime Image Restoration](https://mlsanity.com/paper/2505.16479/) (2025)
- [Efficient Image Restoration with State-Dependent Forward Diffusion](https://mlsanity.com/paper/2505.16733/) (2025)
- [A Unified and Fast-Sampling Diffusion Bridge Framework via Stochastic Optimal Control](https://mlsanity.com/paper/2505.21528/) (2025)
- [Vision Bridge Transformer at Scale](https://mlsanity.com/paper/2511.23199/) (2025)
- [Auditing Sybil: Explaining Deep Lung Cancer Risk Prediction Through Generative Interventional Attributions](https://mlsanity.com/paper/2602.02560/) (2026)
- [Physics-aligned Schr"odinger bridge](https://mlsanity.com/paper/2409.17825/) (2024)
- [DCDB: Dynamic Conditional Dual Diffusion Bridge for Ill-posed Multi-Tasks](https://mlsanity.com/paper/2509.03044/) (2025)
- [Clear Roads, Clear Vision: Advancements in Multi-Weather Restoration for Smart Transportation](https://mlsanity.com/paper/2510.09228/) (2025)
- [Energy-oriented Diffusion Bridge for Image Restoration with Foundational Diffusion Models](https://mlsanity.com/paper/2604.10983/) (2026)
- [Local Intrinsic Dimension Unveils Hallucinations in Diffusion Models](https://mlsanity.com/paper/2605.05026/) (2026)
- [Exploring Diffusion with Test-Time Training on Efficient Image Restoration](https://mlsanity.com/paper/2506.14541/) (2025)
- [Neural Bridge Processes](https://mlsanity.com/paper/2508.07220/) (2025)

## Abstract (arXiv)

Diffusion models exhibit powerful generative capabilities enabling noise mapping to data via reverse stochastic differential equations. However, in image restoration, the focus is on the mapping relationship from low-quality to high-quality images. Regarding this issue, we introduce the Generalized Ornstein-Uhlenbeck Bridge (GOUB) model. By leveraging the natural mean-reverting property of the generalized OU process and further eliminating the variance of its steady-state distribution through the Doob's h-transform, we achieve diffusion mappings from point to point enabling the recovery of high-quality images from low-quality ones. Moreover, we unravel the fundamental mathematical essence shared by various bridge models, all of which are special instances of GOUB and empirically demonstrate the optimality of our proposed models. Additionally, we present the corresponding Mean-ODE model adept at capturing both pixel-level details and structural perceptions. Experimental outcomes showcase the state-of-the-art performance achieved by both models across diverse tasks, including inpainting, deraining, and super-resolution. Code is available at urlhttps://github.com/Hammour-steak/GOUB.