LieWarper: Geometry-Aware Motion Transfer via Lie Algebra

Video motion transfer aims to synthesize novel content videos that strictly follow the motion trajectories of a reference video. However, existing methods typically operate in Euclidean space, treating motion as unconstrained pixel displacements or linear phase shifts. This simplification frequently causes severe shearing artifacts and perspective collapse under complex camera and object motions. In this work, we present LieWarper, a geometry-aware motion transfer framework that reconceptualizes motion as coordinate evolution on a manifold rather than mere pixel displacement. Specifically, we derive an analytic solver on the $\text{Sim}(2)$ manifold to extract global evolution parameters from noisy optical flow. We then introduce a flow-guided phase modulation mechanism, enabling non-rigid dynamics to undergo coordinate transformation along the evolution path. This approach achieves accurate trajectory transfer while maintaining global geometric integrity. Extensive experiments show that LieWarper significantly outperforms state-of-the-art training-free baselines in both motion fidelity and geometric stability, while maintaining high generation quality.