Can Gaussian rendering be both exact and fast
without relying on lossy splatting? Checkout our 3D-GEER!

Abstract

3D Gaussian Splatting (3DGS) marks a significant milestone in balancing the quality and efficiency of differentiable rendering. However, its high efficiency stems from an approximation of projecting 3D Gaussians onto the image plane as 2D Gaussians, which inherently limits rendering quality—particularly under large Field-of-View (FoV) camera inputs. While several recent works have extended 3DGS to mitigate these approximation errors, none have successfully achieved both exactness and high efficiency simultaneously. In this work, we introduce 3DGEER, an Exact and Efficient Volumetric Gaussian Rendering method. Starting from first principles, we derive a closed-form expression for the density integral along a ray traversing a 3D Gaussian distribution. This formulation enables precise forward rendering with arbitrary camera models and supports gradient-based optimization of 3D Gaussian parameters. To ensure both exactness and real-time performance, we propose an efficient method for computing a tight Particle Bounding Frustum (PBF) for each 3D Gaussian, enabling accurate and efficient ray-Gaussian association. We also introduce a novel Bipolar Equiangular Projection (BEAP) representation to accelerate ray association under generic camera models. BEAP further provides a more uniform ray sampling strategy to apply supervision, which empirically improves reconstruction quality. Experiments on multiple pinhole and fisheye datasets show that our method consistently outperforms prior methods, establishing a new state-of-the-art in real-time neural rendering.

Paper Fast Forward

In summary, our contributions are as follows: (i) We present the first complete, first-principle derived, closed-form solution for exact volumetric Gaussian rendering; (ii) We propose an exact and efficient ray-particle association method that enables high speed rendering; (iii) We introduce an equiangular ray sampling strategy that provides more spatially uniform color supervision; (iv) Our method consistently outperforms SOTA approaches across multiple datasets, with notable gains on wide-FoV neural reconstruction benchmarks.