VFOM-GS: Volumetric Fourier Opacity Map for Relightable Gaussian Smoke Reconstruction

Reconstructing dynamic smoke from rendered multi-view videos requires recovering fine-scale geometry while accounting for self-shadowing. Although volumetric Gaussian reconstruction has advanced view synthesis, approaches focused on camera-side transmittance neglect light-side attenuation, baking shadows into primitive colors and limiting relighting. We present VFOM-GS, a shadow-aware 4D Gaussian reconstruction framework that explicitly couples self-shadow formation to the reconstructed density field. Our Volumetric Fourier Opacity Map (VFOM) encodes analytic Gaussian ray profiles in a compact Fourier basis to estimate light-side transmittance. Additionally, the proposed Per-Pixel Depth Normalization (PDN) fits the encoding interval to each light ray's local depth range, improving effective depth resolution. To enable gradient-based optimization of Gaussian parameters, we make both the light and camera passes of our VFOM-based rendering pipeline differentiable. This pipeline models attenuation separately from color while retaining efficient Gaussian rasterization. Experiments on synthetic smoke show that VFOM-GS outperforms the evaluated state-of-the-art baselines in novel-view reconstruction quality and produces consistent self-shadowing under changing illumination. Our Unreal Engine experiments further show that our renderer is nearly twice as fast as the established Niagara system.