Optimizing 3D Gaussian Splatting using Levy Flight

This study investigates the impact of ellipsoid distribution shapes on the rendering performance of 3D Gaussian Splatting (3DGS). We demonstrate, for the first time, that during tile\u2011based rendering, the depth\u2011wise trajectories of Gaussian ellipsoid centers exhibit L{\\'e}vy flight characteristics, driven by the combined effects of the scene\u2019s multi\u2011scale structure and each tile\u2019s spatial perceptual constraints. Building on this insight, we incorporate power\u2011law distributions into our performance analysis and develop a series of ellipsoid functions with varying peak widths and tail thicknesses. Our experimental results show that: In the early training phase (7K iterations), Broad\u2011peaked, thin\u2011tailed ellipsoids accelerate global structure capture and improve all evaluation metrics relative to the standard Gaussian ellipsoid, achieving a 10.7$\\%$ LPIPS improvement on the Mip\u2011NeRF360 dataset. In the late training phase (30K iterations), Sharp\u2011peaked, thin\u2011tailed ellipsoids significantly enhance fine\u2011detail reconstruction, yielding a 17.7$\\%$ LPIPS improvement on the same dataset. This work establishes a novel connection between ellipsoid morphology and rendering performance, systematically characterizing the trade\u2011off between convergence speed and detail fidelity. It offers new theoretical insights and practical guidance for 3DGS and other distribution\u2011based rendering methods.