Geodesic Flow Matching on a Riemannian Degradation Manifold for Blind Image Restoration
For blind image restoration, this work provides a theoretically grounded method to handle complex, mixed degradations by respecting the geometry of degradation space.
This paper introduces a geodesic flow matching framework for blind image restoration that models degradations on a Riemannian manifold, enabling principled handling of mixed and unseen degradations with improved generalization over Euclidean-based methods.
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.