# LinPrim: Linear Primitives for Differentiable Volumetric Rendering

arXiv: https://arxiv.org/abs/2501.16312 · HTML: https://mlsanity.com/paper/2501.16312/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 35 references that are in the map.
- First posted on arXiv 27 Jan 2025.

## Builds on (most cited first, 26 of 26)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [3D Gaussian Splatting for Real-Time Radiance Field Rendering](https://mlsanity.com/paper/2308.04079/) (2023)
- [Instant Neural Graphics Primitives with a Multiresolution Hash Encoding](https://mlsanity.com/paper/2201.05989/) (2022)
- [Mip-NeRF 360: Unbounded Anti-Aliased Neural Radiance Fields](https://mlsanity.com/paper/2111.12077/) (2021)
- [Mip-NeRF: A Multiscale Representation for Anti-Aliasing Neural Radiance Fields](https://mlsanity.com/paper/2103.13415/) (2021)
- [NeuS: Learning Neural Implicit Surfaces by Volume Rendering for Multi-view Reconstruction](https://mlsanity.com/paper/2106.10689/) (2021)
- [D-NeRF: Neural Radiance Fields for Dynamic Scenes](https://mlsanity.com/paper/2011.13961/) (2020)
- [2D Gaussian Splatting for Geometrically Accurate Radiance Fields](https://mlsanity.com/paper/2403.17888/) (2024)
- [DreamGaussian: Generative Gaussian Splatting for Efficient 3D Content Creation](https://mlsanity.com/paper/2309.16653/) (2023)
- [ScanNet++: A High-Fidelity Dataset of 3D Indoor Scenes](https://mlsanity.com/paper/2308.11417/) (2023)
- [Differentiable Volumetric Rendering: Learning Implicit 3D Representations without 3D Supervision](https://mlsanity.com/paper/1912.07372/) (2019)
- [Zip-NeRF: Anti-Aliased Grid-Based Neural Radiance Fields](https://mlsanity.com/paper/2304.06706/) (2023)
- [Mip-Splatting: Alias-free 3D Gaussian Splatting](https://mlsanity.com/paper/2311.16493/) (2023)
- [Point-NeRF: Point-based Neural Radiance Fields](https://mlsanity.com/paper/2201.08845/) (2022)
- [GaussianEditor: Swift and Controllable 3D Editing with Gaussian Splatting](https://mlsanity.com/paper/2311.14521/) (2023)
- [MobileNeRF: Exploiting the Polygon Rasterization Pipeline for Efficient Neural Field Rendering on Mobile Architectures](https://mlsanity.com/paper/2208.00277/) (2022)
- [NeRF-Editing: Geometry Editing of Neural Radiance Fields](https://mlsanity.com/paper/2205.04978/) (2022)
- [3D Gaussian Splatting as Markov Chain Monte Carlo](https://mlsanity.com/paper/2404.09591/) (2024)
- [Mini-Splatting: Representing Scenes with a Constrained Number of Gaussians](https://mlsanity.com/paper/2403.14166/) (2024)
- [Point-Based Neural Rendering with Per-View Optimization](https://mlsanity.com/paper/2109.02369/) (2021)
- [Delicate Textured Mesh Recovery from NeRF via Adaptive Surface Refinement](https://mlsanity.com/paper/2303.02091/) (2023)
- [3D Convex Splatting: Radiance Field Rendering with 3D Smooth Convexes](https://mlsanity.com/paper/2411.14974/) (2024)
- [EVER: Exact Volumetric Ellipsoid Rendering for Real-time View Synthesis](https://mlsanity.com/paper/2410.01804/) (2024)
- [Beyond Gaussians: Fast and High-Fidelity 3D Splatting with Linear Kernels](https://mlsanity.com/paper/2411.12440/) (2024)
- [Efficient Scene Modeling via Structure-Aware and Region-Prioritized 3D Gaussians](https://mlsanity.com/paper/2411.12788/) (2024)
- [Mesh2NeRF: Direct Mesh Supervision for Neural Radiance Field Representation and Generation](https://mlsanity.com/paper/2403.19319/) (2024)

## Built on by (most cited first, 21 of 21)

- [Triangle Splatting for Real-Time Radiance Field Rendering](https://mlsanity.com/paper/2505.19175/) (2025)
- [Faster-GS: Analyzing and Improving Gaussian Splatting Optimization](https://mlsanity.com/paper/2602.09999/) (2026)
- [MeshSplats: Mesh-Based Rendering with Gaussian Splatting Initialization](https://mlsanity.com/paper/2502.07754/) (2025)
- [Radiance Meshes for Volumetric Reconstruction](https://mlsanity.com/paper/2512.04076/) (2025)
- [Splat the Net: Radiance Fields with Splattable Neural Primitives](https://mlsanity.com/paper/2510.08491/) (2025)
- [ConeGS: Error-Guided Densification Using Pixel Cones for Improved Reconstruction With Fewer Primitives](https://mlsanity.com/paper/2511.06810/) (2025)
- [Triangle Splatting+: Differentiable Rendering with Opaque Triangles](https://mlsanity.com/paper/2509.25122/) (2025)
- [ContraGS: Codebook-Condensed and Trainable Gaussian Splatting for Fast, Memory-Efficient Reconstruction](https://mlsanity.com/paper/2509.03775/) (2025)
- [Confidence-Based Mesh Extraction from 3D Gaussians](https://mlsanity.com/paper/2603.24725/) (2026)
- [Radiant Triangle Soup with Soft Connectivity Forces for 3D Reconstruction and Novel View Synthesis](https://mlsanity.com/paper/2505.23642/) (2025)
- [SVR-GS: Spatially Variant Regularization for Probabilistic Masks in 3D Gaussian Splatting](https://mlsanity.com/paper/2509.11116/) (2025)
- [DVSM: Decoder-only View Synthesis Model Done Right](https://mlsanity.com/paper/2605.29891/) (2026)
- [REdiSplats: Ray Tracing for Editable Gaussian Splatting](https://mlsanity.com/paper/2503.12284/) (2025)
- [A Bag of Tricks for Efficient Implicit Neural Point Clouds](https://mlsanity.com/paper/2508.19140/) (2025)
- [Radiant Foam Rendering on a Graph Processor](https://mlsanity.com/paper/2601.04382/) (2026)
- [DiffBMP: Differentiable Rendering with Bitmap Primitives](https://mlsanity.com/paper/2602.22625/) (2026)
- [MLP Splatting: Object-Centric Neural Fields](https://mlsanity.com/paper/2606.03877/) (2026)
- [XPR: An Extensible Cross-Platform Point-Based Differentiable Renderer](https://mlsanity.com/paper/2606.11529/) (2026)
- [Deformable Triangle Splatting: Flexible Primitives for Real-Time Radiance Field Rendering](https://mlsanity.com/paper/2607.22446/) (2026)
- [Differentiable Voronoi Ray Tracing Beyond Rasterization Speeds](https://mlsanity.com/paper/2608.17682/) (2026)
- [Compact Neural Appearance Models for Efficient Gaussian Splatting](https://mlsanity.com/paper/2609.05255/) (2026)

## Abstract (arXiv)

Volumetric rendering has become central to modern novel view synthesis methods, which use differentiable rendering to optimize 3D scene representations directly from observed views. While many recent works build on NeRF or 3D Gaussians, we explore an alternative volumetric scene representation. More specifically, we introduce two new scene representations based on linear primitives - octahedra and tetrahedra - both of which define homogeneous volumes bounded by triangular faces. To optimize these primitives, we present a differentiable rasterizer that runs efficiently on GPUs, allowing end-to-end gradient-based optimization while maintaining real-time rendering capabilities. Through experiments on real-world datasets, we demonstrate comparable performance to state-of-the-art volumetric methods while requiring fewer primitives to achieve similar reconstruction fidelity. Our findings deepen the understanding of 3D representations by providing insights into the fidelity and performance characteristics of transparent polyhedra and suggest that adopting novel primitives can expand the available design space.