Moment-Based 3D Gaussian Splatting: Resolving Volumetric Occlusion with Order-Independent Transmittance

Jan U. Müller, Robin Tim Landsgesell, Leif Van Holland, Patrick Stotko, Reinhard Klein
University of Bonn
Paper > arXiv Video Code
Overview of the MB3DGS pipeline showing the forward pass transition from Gaussian Particles to render image.

Breakdown of the Moment-based 3D Gaussian Splatting render pipeline, illustrating the flow from input primitives to final pixel color:

  1. Gaussian Particles, which represent the scene using explicitly parameterized primitives that define a localized density distribution. Unlike standard 3DGS, these particles are treated as a continuous volumetric medium rather than discrete alpha-blended splats.
  2. Moment Pass, where per-pixel density moments are analytically computed and accumulated from all contributing Gaussians. These moments allow for the reconstruction of a continuous, order-independent transmittance function along each camera ray.
  3. Quadrature Pass, which independently evaluates the volume rendering integral for each Gaussian using the reconstructed transmittance. This pass accumulates the emitted radiance via an efficient numerical quadrature rule derived from piecewise-constant density assumptions.
  4. Rescaling Pass, which normalizes the accumulated radiance using the total optical depth to correct for approximation errors and ensure the final image maintains correct scene opacity.

Abstract

The recent success of 3D Gaussian Splatting (3DGS) has reshaped novel view synthesis by enabling fast optimization and real-time rendering of high-quality radiance fields. However, it relies on simplified, order-dependent alpha blending and coarse approximations of the density integral within the rasterizer, thereby limiting its ability to render complex, overlapping semi-transparent objects. In this paper, we extend rasterization-based rendering of 3D Gaussian representations with a novel method for high-fidelity transmittance computation, entirely avoiding the need for ray tracing or per-pixel sample sorting. Building on prior work in moment-based order-independent transparency, our key idea is to characterize the density distribution along each camera ray with a compact and continuous representation based on statistical moments. To this end, we analytically derive and compute a set of per-pixel moments from all contributing 3D Gaussians. From these moments, a continuous transmittance function is reconstructed for each ray, which is then independently sampled within each Gaussian. As a result, our method bridges the gap between rasterization and physical accuracy by modeling light attenuation in complex translucent media, significantly improving overall reconstruction and rendering quality.

Acknowledgments

This work has been funded by the Federal Ministry of Research, Technology and Space of Germany and the state of North Rhine-Westphalia as part of the Lamarr Institute for Machine Learning and Artificial Intelligence, by the European Regional Development Fund and the state of North Rhine-Westphalia under grant number EFRE-20801085 (Gen-AIvatar), by the state of North Rhine-Westphalia as part of the Excellency Start-up Center.NRW (U-BO-GROW) under grant number 03ESCNW18B, and additionally by the Ministry of Culture and Science North Rhine-Westphalia under grant number PB22-063A (InVirtuo 4.0: Experimental Research in Virtual Environments).

Kofinanziert von der Europäischen Union
IN.NRW INNOVATIONSFÖRDERAGENTUR
Ministerium für Wirtschaft, Industrie, Klimaschutz und Engergie des Landes Nordrhein-Westfalen