# I2SB: Image-to-Image Schr"odinger Bridge

arXiv: https://arxiv.org/abs/2302.05872 · HTML: https://mlsanity.com/paper/2302.05872/

## Key facts

- Cited by 218 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 45 of its 79 references that are in the map.
- First posted on arXiv 12 Feb 2023.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 30 of 45)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [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)
- [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)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [Photo-Realistic Single Image Super-Resolution Using a Generative Adversarial Network](https://mlsanity.com/paper/1609.04802/) (2016)
- [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)
- [Palette: Image-to-Image Diffusion Models](https://mlsanity.com/paper/2111.05826/) (2021)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Transformers in Vision: A Survey](https://mlsanity.com/paper/2101.01169/) (2021)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Multimodal Unsupervised Image-to-Image Translation](https://mlsanity.com/paper/1804.04732/) (2018)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Free-Form Image Inpainting with Gated Convolution](https://mlsanity.com/paper/1806.03589/) (2018)
- [Zero-Shot Image Restoration Using Denoising Diffusion Null-Space Model](https://mlsanity.com/paper/2212.00490/) (2022)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)

## Built on by (most cited first, 30 of 218)

- [ResShift: Efficient Diffusion Model for Image Super-resolution by Residual Shifting](https://mlsanity.com/paper/2307.12348/) (2023)
- [UniControl: A Unified Diffusion Model for Controllable Visual Generation In the Wild](https://mlsanity.com/paper/2305.11147/) (2023)
- [A Survey on Diffusion Models for Inverse Problems](https://mlsanity.com/paper/2410.00083/) (2024)
- [Denoising Diffusion Bridge Models](https://mlsanity.com/paper/2309.16948/) (2023)
- [Fast Training of Diffusion Models with Masked Transformers](https://mlsanity.com/paper/2306.09305/) (2023)
- [Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restoration](https://mlsanity.com/paper/2303.11435/) (2023)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023)
- [Fine-Tuning of Continuous-Time Diffusion Models as Entropy-Regularized Control](https://mlsanity.com/paper/2402.15194/) (2024)
- [CADS: Unleashing the Diversity of Diffusion Models through Condition-Annealed Sampling](https://mlsanity.com/paper/2310.17347/) (2023)
- [Eliminating Oversaturation and Artifacts of High Guidance Scales in Diffusion Models](https://mlsanity.com/paper/2410.02416/) (2024)
- [Unpaired Image-to-Image Translation via Neural Schr"odinger Bridge](https://mlsanity.com/paper/2305.15086/) (2023)
- [Simulation-free Schr"odinger bridges via score and flow matching](https://mlsanity.com/paper/2307.03672/) (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)
- [Residual Denoising Diffusion Models](https://mlsanity.com/paper/2308.13712/) (2023)
- [On the Generalization Properties of Diffusion Models](https://mlsanity.com/paper/2311.01797/) (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)
- [No Training, No Problem: Rethinking Classifier-Free Guidance for Diffusion Models](https://mlsanity.com/paper/2407.02687/) (2024)
- [Diffusion Bridge Implicit Models](https://mlsanity.com/paper/2405.15885/) (2024)
- [DEFT: Efficient Fine-Tuning of Diffusion Models by Learning the Generalised h-transform](https://mlsanity.com/paper/2406.01781/) (2024)
- [Meta Flow Matching: Integrating Vector Fields on the Wasserstein Manifold](https://mlsanity.com/paper/2408.14608/) (2024)
- [Mirror Diffusion Models for Constrained and Watermarked Generation](https://mlsanity.com/paper/2310.01236/) (2023)
- [Image Restoration Through Generalized Ornstein-Uhlenbeck Bridge](https://mlsanity.com/paper/2312.10299/) (2023)
- [Generalized Schr"odinger Bridge Matching](https://mlsanity.com/paper/2310.02233/) (2023)
- [Rethinking Score Distillation as a Bridge Between Image Distributions](https://mlsanity.com/paper/2406.09417/) (2024)
- [LBM: Latent Bridge Matching for Fast Image-to-Image Translation](https://mlsanity.com/paper/2503.07535/) (2025)
- [FlowTok: Flowing Seamlessly Across Text and Image Tokens](https://mlsanity.com/paper/2503.10772/) (2025)
- [Accurate transition state generation with an object-aware equivariant elementary reaction diffusion model](https://mlsanity.com/paper/2304.06174/) (2023)

## Abstract (arXiv)

We propose Image-to-Image Schr"odinger Bridge (I2SB), a new class of conditional diffusion models that directly learn the nonlinear diffusion processes between two given distributions. These diffusion bridges are particularly useful for image restoration, as the degraded images are structurally informative priors for reconstructing the clean images. I2SB belongs to a tractable class of Schr"odinger bridge, the nonlinear extension to score-based models, whose marginal distributions can be computed analytically given boundary pairs. This results in a simulation-free framework for nonlinear diffusions, where the I2SB training becomes scalable by adopting practical techniques used in standard diffusion models. We validate I2SB in solving various image restoration tasks, including inpainting, super-resolution, deblurring, and JPEG restoration on ImageNet 256x256 and show that I2SB surpasses standard conditional diffusion models with more interpretable generative processes. Moreover, I2SB matches the performance of inverse methods that additionally require the knowledge of the corruption operators. Our work opens up new algorithmic opportunities for developing efficient nonlinear diffusion models on a large scale. scale. Project page and codes: https://i2sb.github.io/