# Geodesic Flow Matching on a Riemannian Degradation Manifold for Blind Image Restoration

arXiv: https://arxiv.org/abs/2606.06278 · HTML: https://mlsanity.com/paper/2606.06278/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 22 of its 35 references that are in the map.
- First posted on arXiv 4 Jun 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 22 of 22)

- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Adding Conditional Control to Text-to-Image Diffusion Models](https://mlsanity.com/paper/2302.05543/) (2023)
- [InstructPix2Pix: Learning to Follow Image Editing Instructions](https://mlsanity.com/paper/2211.09800/) (2022)
- [Restormer: Efficient Transformer for High-Resolution Image Restoration](https://mlsanity.com/paper/2111.09881/) (2021)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [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)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [TransWeather: Transformer-based Restoration of Images Degraded by Adverse Weather Conditions](https://mlsanity.com/paper/2111.14813/) (2021)
- [DesnowNet: Context-Aware Deep Network for Snow Removal](https://mlsanity.com/paper/1708.04512/) (2017)
- [Heavy Rain Image Restoration: Integrating Physics Model and Conditional Adversarial Learning](https://mlsanity.com/paper/1904.05050/) (2019)
- [Dense Haze: A benchmark for image dehazing with dense-haze and haze-free images](https://mlsanity.com/paper/1904.02904/) (2019)
- [Diff-Plugin: Revitalizing Details for Diffusion-based Low-level Tasks](https://mlsanity.com/paper/2403.00644/) (2024)
- [From Sky to the Ground: A Large-scale Benchmark and Simple Baseline Towards Real Rain Removal](https://mlsanity.com/paper/2308.03867/) (2023)
- [LBM: Latent Bridge Matching for Fast Image-to-Image Translation](https://mlsanity.com/paper/2503.07535/) (2025)
- [Towards Unified Deep Image Deraining: A Survey and A New Benchmark](https://mlsanity.com/paper/2310.03535/) (2023)
- [Reversing Flow for Image Restoration](https://mlsanity.com/paper/2506.16961/) (2025)

## Abstract (arXiv)

Blind image restoration requires recovering clean images from observations corrupted by unknown and potentially mixed degradations. While recent deterministic flow-based methods model restoration as transport processes that map degraded images to clean ones, they typically rely on Euclidean interpolation, implicitly assuming linear degradation geometry. In this paper, we explicitly model degradations as points on a low-dimensional Riemannian manifold and formulate restoration as geodesic transport on the joint image-manifold space. Using a geodesic flow matching objective, we learn intrinsic transport dynamics that respect the curvature of degradation space. This framework generalizes linear flow matching, provides a principled treatment of mixed degradations as geodesic compositions, and yields a clean theoretical interpretation for generalization beyond observed degradations.