Eulerian Gaussian Splatting using Hashed Probability Pyramids

We introduce a probabilistic splat-based radiance field framework that retains the fast rasterization and test-time efficiency of 3D Gaussian Splatting (3DGS) while replacing heuristic primitive manipulation with gradient-based optimization of a volumetric probability density. Rather than relocating, splitting, or culling Gaussians via hand-tuned densification (e.g., ADC), we treat primitive locations as samples drawn from a persistent, learnable density. We instantiate this density with a novel, memory-efficient multi-scale hierarchical grid that enables end-to-end gradient-based control over primitive population density. To stabilize stochastic training, we derive an unbiased gradient estimator with control variates that markedly reduces variance. By allowing probability mass to flow to where the loss demands, our method eliminates brittle priors and naturally explores the volume, achieving state-of-the-art reconstruction quality on mip-NeRF360 while preserving 3DGS-level rendering speed.