GaussianSpline: Adaptive Bézier Splines for Highly Dynamic 4D Gaussian Splatting

We present GaussianSpline, a dynamic 4D Gaussian Splatting method that models per-Gaussian motion using adaptive, non-uniform piecewise cubic Bézier splines. Unlike approaches based on global deformations, fixed-order polynomials, or uniformly parameterized splines, GaussianSpline jointly learns the Bézier control points and the temporal partition of each Gaussian's trajectory. This adaptive representation allocates greater temporal capacity to highly dynamic regions while maintaining compact trajectories for near-static regions, enabling high-fidelity reconstruction of complex and rapidly changing motion. To jointly achieve globally smooth motion and local adaptivity, we enforce $G^1$ continuity between consecutive Bézier segments using a reflection-based constraint on their control points. This produces smooth, fully differentiable trajectories while preserving the flexibility of non-uniform temporal parameterization. Experiments on the Neural 3D Video benchmark, the Google Immersive dataset, as well as on synthetic scenes demonstrate that GaussianSpline substantially improves reconstruction quality in scenes with highly non-uniform and challenging dynamics, while maintaining competitive performance on general dynamic scenes compared with state-of-the-art 4D Gaussian Splatting methods.