# Posterior-Mean Rectified Flow: Towards Minimum MSE Photo-Realistic Image Restoration

arXiv: https://arxiv.org/abs/2410.00418 · HTML: https://mlsanity.com/paper/2410.00418/

## Key facts

- Cited by 34 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 66 references that are in the map.
- First posted on arXiv 1 Oct 2024.

## 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)

- [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)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [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)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Photo-Realistic Single Image Super-Resolution Using a Generative Adversarial Network](https://mlsanity.com/paper/1609.04802/) (2016)
- [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)
- [Palette: Image-to-Image Diffusion Models](https://mlsanity.com/paper/2111.05826/) (2021)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [ESRGAN: Enhanced Super-Resolution Generative Adversarial Networks](https://mlsanity.com/paper/1809.00219/) (2018)
- [Colorful Image Colorization](https://mlsanity.com/paper/1603.08511/) (2016)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Real-ESRGAN: Training Real-World Blind Super-Resolution with Pure Synthetic Data](https://mlsanity.com/paper/2107.10833/) (2021)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Improved Precision and Recall Metric for Assessing Generative Models](https://mlsanity.com/paper/1904.06991/) (2019)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [The Perception-Distortion Tradeoff](https://mlsanity.com/paper/1711.06077/) (2017)
- [Zero-Shot Image Restoration Using Denoising Diffusion Null-Space Model](https://mlsanity.com/paper/2212.00490/) (2022)
- [Designing a Practical Degradation Model for Deep Blind Image Super-Resolution](https://mlsanity.com/paper/2103.14006/) (2021)
- [NIMA: Neural Image Assessment](https://mlsanity.com/paper/1709.05424/) (2017)
- [Towards Real-World Blind Face Restoration with Generative Facial Prior](https://mlsanity.com/paper/2101.04061/) (2021)
- [Denoising Diffusion Models for Plug-and-Play Image Restoration](https://mlsanity.com/paper/2305.08995/) (2023)
- [DiffBIR: Towards Blind Image Restoration with Generative Diffusion Prior](https://mlsanity.com/paper/2308.15070/) (2023)
- [Towards Robust Blind Face Restoration with Codebook Lookup Transformer](https://mlsanity.com/paper/2206.11253/) (2022)
- [BBDM: Image-to-image Translation with Brownian Bridge Diffusion Models](https://mlsanity.com/paper/2205.07680/) (2022)

## Built on by (most cited first, 30 of 34)

- [Compressed Image Generation with Denoising Diffusion Codebook Models](https://mlsanity.com/paper/2502.01189/) (2025)
- [Efficient Image Restoration via Latent Consistency Flow Matching](https://mlsanity.com/paper/2502.03500/) (2025)
- [Low Light Image Enhancement Challenge at NTIRE 2026](https://mlsanity.com/paper/2604.17669/) (2026)
- [Kernel Density Steering: Inference-Time Scaling via Mode Seeking for Image Restoration](https://mlsanity.com/paper/2507.05604/) (2025)
- [Prior-Informed Flow Matching for Graph Reconstruction](https://mlsanity.com/paper/2601.22107/) (2026)
- [HazeMatching: Dehazing Light Microscopy Images with Guided Conditional Flow Matching](https://mlsanity.com/paper/2506.22397/) (2025)
- [Diffusion Bridge or Flow Matching? A Unifying Framework and Comparative Analysis](https://mlsanity.com/paper/2509.24531/) (2025)
- [Reconstruct Anything Model: a lightweight general model for computational imaging](https://mlsanity.com/paper/2503.08915/) (2025)
- [Efficient Image Restoration with State-Dependent Forward Diffusion](https://mlsanity.com/paper/2505.16733/) (2025)
- [Flow Diverse and Efficient: Learning Momentum Flow Matching via Stochastic Velocity Field Sampling](https://mlsanity.com/paper/2506.08796/) (2025)
- [Unlocking the Potential of Diffusion Priors in Blind Face Restoration](https://mlsanity.com/paper/2508.08556/) (2025)
- [Contrast-X: A Multi-Modal Contrast Image Synthesis Benchmark and Universal Modality Flow Matching](https://mlsanity.com/paper/2601.15884/) (2026)
- [A Simple Combination of Diffusion Models for Better Quality Trade-Offs in Image Denoising](https://mlsanity.com/paper/2503.14654/) (2025)
- [V.I.P. : Iterative Online Preference Distillation for Efficient Video Diffusion Models](https://mlsanity.com/paper/2508.03254/) (2025)
- [Synthesizing Accurate and Realistic T1-weighted Contrast-Enhanced MR Images using Posterior-Mean Rectified Flow](https://mlsanity.com/paper/2508.12640/) (2025)
- [Regularized Schr"odinger Bridge via Distortion-Perception Perturbation for High-Fidelity Speech Enhancement](https://mlsanity.com/paper/2511.11686/) (2025)
- [Categorical Reparameterization with Denoising Diffusion models](https://mlsanity.com/paper/2601.00781/) (2026)
- [SP3: Spherical Priors for Plug-and-Play Restoration](https://mlsanity.com/paper/2606.16396/) (2026)
- [Flow Straight to Reality: Perceptually Consistent Flow Matching for Efficient Image Restoration](https://mlsanity.com/paper/2608.10544/) (2026)
- [End-to-End PET Image Reconstruction via a Posterior-Mean Diffusion Model](https://mlsanity.com/paper/2503.08546/) (2025)
- [On Inverse Problems, Parameter Estimation, and Domain Generalization](https://mlsanity.com/paper/2506.06024/) (2025)
- [Latent Posterior-Mean Rectified Flow for Higher-Fidelity Perceptual Face Restoration](https://mlsanity.com/paper/2507.00447/) (2025)
- [Unsupervised Real-World Super-Resolution via Rectified Flow Degradation Modelling](https://mlsanity.com/paper/2508.07214/) (2025)
- [Optimizing ID Consistency in Multimodal Large Models: Facial Restoration via Alignment, Entanglement, and Disentanglement](https://mlsanity.com/paper/2602.18752/) (2026)
- [Linearized Coupling Flow with Shortcut Constraints for One-Step Face Restoration](https://mlsanity.com/paper/2603.03648/) (2026)
- [IR-Flow: Bridging Discriminative and Generative Image Restoration via Rectified Flow](https://mlsanity.com/paper/2604.19680/) (2026)
- [Flow-based Gaussian Splatting for Continuous-Scale Remote Sensing Image Super-Resolution](https://mlsanity.com/paper/2605.22147/) (2026)
- [Resolving Endpoint Underfitting in Diffusion Bridges via Noise Alignment](https://mlsanity.com/paper/2605.28962/) (2026)
- [DRIFT: Difficulty-aware Rectified Flows for Through-plane MRI Super-Resolution](https://mlsanity.com/paper/2607.16649/) (2026)
- [Fast and Faithful: Principled Conditional Flow Matching for Inverse Problems](https://mlsanity.com/paper/2609.12953/) (2026)

## Abstract (arXiv)

Photo-realistic image restoration algorithms are typically evaluated by distortion measures (e.g., PSNR, SSIM) and by perceptual quality measures (e.g., FID, NIQE), where the desire is to attain the lowest possible distortion without compromising on perceptual quality. To achieve this goal, current methods commonly attempt to sample from the posterior distribution, or to optimize a weighted sum of a distortion loss (e.g., MSE) and a perceptual quality loss (e.g., GAN). Unlike previous works, this paper is concerned specifically with the optimal estimator that minimizes the MSE under a constraint of perfect perceptual index, namely where the distribution of the reconstructed images is equal to that of the ground-truth ones. A recent theoretical result shows that such an estimator can be constructed by optimally transporting the posterior mean prediction (MMSE estimate) to the distribution of the ground-truth images. Inspired by this result, we introduce Posterior-Mean Rectified Flow (PMRF), a simple yet highly effective algorithm that approximates this optimal estimator. In particular, PMRF first predicts the posterior mean, and then transports the result to a high-quality image using a rectified flow model that approximates the desired optimal transport map. We investigate the theoretical utility of PMRF and demonstrate that it consistently outperforms previous methods on a variety of image restoration tasks.