# NeRF Revisited: Fixing Quadrature Instability in Volume Rendering

arXiv: https://arxiv.org/abs/2310.20685 · HTML: https://mlsanity.com/paper/2310.20685/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 37 references that are in the map.
- First posted on arXiv 31 Oct 2023.

## Builds on (most cited first, 25 of 25)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [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)
- [pixelNeRF: Neural Radiance Fields from One or Few Images](https://mlsanity.com/paper/2012.02190/) (2020)
- [NeRF in the Wild: Neural Radiance Fields for Unconstrained Photo Collections](https://mlsanity.com/paper/2008.02268/) (2020)
- [Nerfies: Deformable Neural Radiance Fields](https://mlsanity.com/paper/2011.12948/) (2020)
- [Direct Voxel Grid Optimization: Super-fast Convergence for Radiance Fields Reconstruction](https://mlsanity.com/paper/2111.11215/) (2021)
- [Neural Sparse Voxel Fields](https://mlsanity.com/paper/2007.11571/) (2020)
- [Neural Scene Flow Fields for Space-Time View Synthesis of Dynamic Scenes](https://mlsanity.com/paper/2011.13084/) (2020)
- [Depth-supervised NeRF: Fewer Views and Faster Training for Free](https://mlsanity.com/paper/2107.02791/) (2021)
- [Block-NeRF: Scalable Large Scene Neural View Synthesis](https://mlsanity.com/paper/2202.05263/) (2022)
- [IBRNet: Learning Multi-View Image-Based Rendering](https://mlsanity.com/paper/2102.13090/) (2021)
- [Local Light Field Fusion: Practical View Synthesis with Prescriptive Sampling Guidelines](https://mlsanity.com/paper/1905.00889/) (2019)
- [Zip-NeRF: Anti-Aliased Grid-Based Neural Radiance Fields](https://mlsanity.com/paper/2304.06706/) (2023)
- [Nerfstudio: A Modular Framework for Neural Radiance Field Development](https://mlsanity.com/paper/2302.04264/) (2023)
- [Baking Neural Radiance Fields for Real-Time View Synthesis](https://mlsanity.com/paper/2103.14645/) (2021)
- [NeRF in the Dark: High Dynamic Range View Synthesis from Noisy Raw Images](https://mlsanity.com/paper/2111.13679/) (2021)
- [MobileNeRF: Exploiting the Polygon Rasterization Pipeline for Efficient Neural Field Rendering on Mobile Architectures](https://mlsanity.com/paper/2208.00277/) (2022)
- [BungeeNeRF: Progressive Neural Radiance Field for Extreme Multi-scale Scene Rendering](https://mlsanity.com/paper/2112.05504/) (2021)
- [Differentiable Rendering: A Survey](https://mlsanity.com/paper/2006.12057/) (2020)
- [Improved Direct Voxel Grid Optimization for Radiance Fields Reconstruction](https://mlsanity.com/paper/2206.05085/) (2022)
- [DIVeR: Real-time and Accurate Neural Radiance Fields with Deterministic Integration for Volume Rendering](https://mlsanity.com/paper/2111.10427/) (2021)
- [SCADE: NeRFs from Space Carving with Ambiguity-Aware Depth Estimates](https://mlsanity.com/paper/2303.13582/) (2023)

## Built on by (most cited first, 10 of 10)

- [Neural Visibility Field for Uncertainty-Driven Active Mapping](https://mlsanity.com/paper/2406.06948/) (2024)
- [Neural Radiance Fields for the Real World: A Survey](https://mlsanity.com/paper/2501.13104/) (2025)
- [Volumetric Rendering with Baked Quadrature Fields](https://mlsanity.com/paper/2312.02202/) (2023)
- [L0-Sampler: An L0 Model Guided Volume Sampling for NeRF](https://mlsanity.com/paper/2311.07044/) (2023)
- [A Unified Theory of Sinusoidal Activation Families for Implicit Neural Representations](https://mlsanity.com/paper/2502.00869/) (2025)
- [ProvNeRF: Modeling per Point Provenance in NeRFs as a Stochastic Field](https://mlsanity.com/paper/2401.08140/) (2024)
- [GL-NeRF: Gauss-Laguerre Quadrature Enables Training-Free NeRF Acceleration](https://mlsanity.com/paper/2410.19831/) (2024)
- [Ternary-Type Opacity and Hybrid Odometry for RGB NeRF-SLAM](https://mlsanity.com/paper/2312.13332/) (2023)
- [Evolutive Rendering Models](https://mlsanity.com/paper/2405.17531/) (2024)
- [Deformable Triangle Splatting: Flexible Primitives for Real-Time Radiance Field Rendering](https://mlsanity.com/paper/2607.22446/) (2026)

## Abstract (arXiv)

Neural radiance fields (NeRF) rely on volume rendering to synthesize novel views. Volume rendering requires evaluating an integral along each ray, which is numerically approximated with a finite sum that corresponds to the exact integral along the ray under piecewise constant volume density. As a consequence, the rendered result is unstable w.r.t. the choice of samples along the ray, a phenomenon that we dub quadrature instability. We propose a mathematically principled solution by reformulating the sample-based rendering equation so that it corresponds to the exact integral under piecewise linear volume density. This simultaneously resolves multiple issues: conflicts between samples along different rays, imprecise hierarchical sampling, and non-differentiability of quantiles of ray termination distances w.r.t. model parameters. We demonstrate several benefits over the classical sample-based rendering equation, such as sharper textures, better geometric reconstruction, and stronger depth supervision. Our proposed formulation can be also be used as a drop-in replacement to the volume rendering equation of existing NeRF-based methods. Our project page can be found at pl-nerf.github.io.