Exact-GS: Mathematically Rigorous and Accurate 3D Gaussian Splatting for 3D X-ray Reconstruction

We propose Exact-GS, a novel mathematically rigorous and accurate 3D Gaussian Splatting model designed to perform 3D X-ray computed tomography (CT) reconstruction and novel view synthesis. Recently, 3D Gaussian Splatting achieved considerable progress at 3D representation. Unfortunately, due to the affine approximation of the projective transformation, previous 3DGS-based methods inevitably suffer from artifacts and projection inconsistencies. To address this problem, some ray tracing based methods perform integration along the ray across Gaussians. However, these methods are computationally inefficient on the forward and backward pass. We introduce a novel closed-form splatting solution for this problem with mathematically rigorous derivation. Our model is the first to achieve the same exact rendering quality as ray tracing based methods without any approximation under a splatting-based formulation, enabling fast CUDA-based hardware rasterization.Additionally, we present a precise Gaussian-tile intersection algorithm, enabling faster and efficient rendering.We demonstrate the performance gains by reconstruction and novel view synthesis through different synthetic and real-world datasets. Our method also contributes to the visible light scene representation, where the density accumulation (X-ray attenuation coefficient) in our model can be replaced by the integral of the opacity at alpha blending.