Geometry-Aware Image Flow Matching

2026-05-24

AI Summary

Natural image generation typically relies on Euclidean assumptions, overlooking the intrinsic nonlinear geometric structure of the data. This work observes that image semantics are primarily governed by directional components, while the magnitude can be approximated by a global mean, enabling images to be modeled on a hyperspherical manifold. Building on this insight, the paper introduces two novel approaches: Spherical Optimal Transport Conditional Flow Matching (SOT-CFM), which leverages angular distances, and Spherical Flow Matching (SFM), which directly constrains the dynamics to the manifold. To the best of our knowledge, this is the first successful formulation of natural image generation on a hypersphere via flow matching. Experiments demonstrate that the proposed methods significantly outperform Euclidean baselines across multiple benchmarks, confirming the advantages of geometry-aware modeling in generative tasks.

Abstract

Recent advances in generative models highlight the power of geometry-aware modeling in manifold-constrained settings. Yet, for natural images, the field remains confined to Euclidean assumptions, failing to exploit the potential of intrinsic geometric structures within the data. In this work, we investigate the geometry of natural images and observe that semantic information is predominantly encoded in directional components, while norm components can be approximated by the global average. This property holds across both RGB and latent spaces, suggesting that natural images can be effectively modeled on a hypersphere. Building on this finding, we introduce Spherical Optimal Transport Flow Matching (SOT-CFM), which utilizes angular distance, and Spherical Flow Matching (SFM), which constrains dynamics directly on the manifold. Our experiments demonstrate that these geometry-aware methods achieve superior performance against Euclidean baselines. Ultimately, this work provides a novel perspective that bridges the gap between Riemannian manifold-based modeling and natural image generation.

Problem

• geometry-aware modeling
• natural images
• manifold constraints
• Riemannian manifold
• image generation

Innovation

• geometry-aware modeling
• spherical flow matching
• manifold-constrained generation
• angular distance
• natural image generation