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

2026-06-04

AI Summary

This work addresses the challenge of blind image restoration under unknown and potentially mixed degradation processes, which traditional methods struggle to model due to their reliance on Euclidean space assumptions that fail to capture the intrinsic nonlinear geometric structure. The paper introduces a novel framework that explicitly models degradation as points on a low-dimensional Riemannian manifold and incorporates a geodesic transport mechanism within the joint space of images and degradation manifolds. By learning restoration dynamics through geodesic flow matching that respects the manifold’s curvature, the method unifies the handling of mixed degradations via geodesic interpolation. This geometrically principled approach yields a nonlinear restoration framework more aligned with the true nature of degradation, significantly improving both restoration performance and generalization under complex and unseen degradation scenarios, with accompanying theoretical guarantees.

Abstract

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.

Problem

• blind image restoration
• degradation manifold
• geodesic flow
• Riemannian geometry
• mixed degradations

Innovation

• Geodesic Flow Matching
• Riemannian Manifold
• Blind Image Restoration
• Degradation Modeling
• Transport Dynamics