# Exact-NeRF: An Exploration of a Precise Volumetric Parameterization for Neural Radiance Fields

arXiv: https://arxiv.org/abs/2211.12285 · HTML: https://mlsanity.com/paper/2211.12285/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 40 references that are in the map.
- First posted on arXiv 22 Nov 2022.

## Builds on (most cited first, 30 of 30)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [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)
- [Plenoxels: Radiance Fields without Neural Networks](https://mlsanity.com/paper/2112.05131/) (2021)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [NeRF in the Wild: Neural Radiance Fields for Unconstrained Photo Collections](https://mlsanity.com/paper/2008.02268/) (2020)
- [PlenOctrees for Real-time Rendering of Neural Radiance Fields](https://mlsanity.com/paper/2103.14024/) (2021)
- [NeRF++: Analyzing and Improving Neural Radiance Fields](https://mlsanity.com/paper/2010.07492/) (2020)
- [Image Quality Assessment: Unifying Structure and Texture Similarity](https://mlsanity.com/paper/2004.07728/) (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)
- [BARF: Bundle-Adjusting Neural Radiance Fields](https://mlsanity.com/paper/2104.06405/) (2021)
- [RegNeRF: Regularizing Neural Radiance Fields for View Synthesis from Sparse Inputs](https://mlsanity.com/paper/2112.00724/) (2021)
- [Ref-NeRF: Structured View-Dependent Appearance for Neural Radiance Fields](https://mlsanity.com/paper/2112.03907/) (2021)
- [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)
- [Light Field Networks: Neural Scene Representations with Single-Evaluation Rendering](https://mlsanity.com/paper/2106.02634/) (2021)
- [Urban Radiance Fields](https://mlsanity.com/paper/2111.14643/) (2021)
- [DONeRF: Towards Real-Time Rendering of Compact Neural Radiance Fields using Depth Oracle Networks](https://mlsanity.com/paper/2103.03231/) (2021)
- [BungeeNeRF: Progressive Neural Radiance Field for Extreme Multi-scale Scene Rendering](https://mlsanity.com/paper/2112.05504/) (2021)
- [Deblur-NeRF: Neural Radiance Fields from Blurry Images](https://mlsanity.com/paper/2111.14292/) (2021)
- [HDR-NeRF: High Dynamic Range Neural Radiance Fields](https://mlsanity.com/paper/2111.14451/) (2021)
- [EfficientNeRF: Efficient Neural Radiance Fields](https://mlsanity.com/paper/2206.00878/) (2022)
- [MedNeRF: Medical Neural Radiance Fields for Reconstructing 3D-aware CT-Projections from a Single X-ray](https://mlsanity.com/paper/2202.01020/) (2022)
- [On the Suitability of Lp-norms for Creating and Preventing Adversarial Examples](https://mlsanity.com/paper/1802.09653/) (2018)
- [NeuroFluid: Fluid Dynamics Grounding with Particle-Driven Neural Radiance Fields](https://mlsanity.com/paper/2203.01762/) (2022)
- [Recursive-NeRF: An Efficient and Dynamically Growing NeRF](https://mlsanity.com/paper/2105.09103/) (2021)
- [Gravitationally Lensed Black Hole Emission Tomography](https://mlsanity.com/paper/2204.03715/) (2022)
- [Mip-NeRF RGB-D: Depth Assisted Fast Neural Radiance Fields](https://mlsanity.com/paper/2205.09351/) (2022)

## Built on by (most cited first, 15 of 15)

- [PyNeRF: Pyramidal Neural Radiance Fields](https://mlsanity.com/paper/2312.00252/) (2023)
- [Neural Radiance Field-based Visual Rendering: A Comprehensive Review](https://mlsanity.com/paper/2404.00714/) (2024)
- [Neural Field Convolutions by Repeated Differentiation](https://mlsanity.com/paper/2304.01834/) (2023)
- [InsertNeRF: Instilling Generalizability into NeRF with HyperNet Modules](https://mlsanity.com/paper/2308.13897/) (2023)
- [TraM-NeRF: Tracing Mirror and Near-Perfect Specular Reflections through Neural Radiance Fields](https://mlsanity.com/paper/2310.10650/) (2023)
- [Fast Sun-aligned Outdoor Scene Relighting based on TensoRF](https://mlsanity.com/paper/2311.03965/) (2023)
- [Hyb-NeRF: A Multiresolution Hybrid Encoding for Neural Radiance Fields](https://mlsanity.com/paper/2311.12490/) (2023)
- [LookCloser: Frequency-aware Radiance Field for Tiny-Detail Scene](https://mlsanity.com/paper/2503.18513/) (2025)
- [Anisotropic Neural Representation Learning for High-Quality Neural Rendering](https://mlsanity.com/paper/2311.18311/) (2023)
- [MomentsNeRF: Leveraging Orthogonal Moments for Few-Shot Neural Rendering](https://mlsanity.com/paper/2407.02668/) (2024)
- [Deep 3D World Models for Multi-Image Super-Resolution Beyond Optical Flow](https://mlsanity.com/paper/2401.16972/) (2024)
- [ARC-NeRF: Area Ray Casting for Broader Unseen View Coverage in Few-shot Object Rendering](https://mlsanity.com/paper/2403.10906/) (2024)
- [Objectness Similarity: Capturing Object-Level Fidelity in 3D Scene Evaluation](https://mlsanity.com/paper/2509.09143/) (2025)
- [ImmersiveNeRF: Hybrid Radiance Fields for Unbounded Immersive Light Field Reconstruction](https://mlsanity.com/paper/2309.01374/) (2023)
- [Aperture: Merge-Consistent Rotary States for Compressed Tokens](https://mlsanity.com/paper/2609.36781/) (2026)

## Abstract (arXiv)

Neural Radiance Fields (NeRF) have attracted significant attention due to their ability to synthesize novel scene views with great accuracy. However, inherent to their underlying formulation, the sampling of points along a ray with zero width may result in ambiguous representations that lead to further rendering artifacts such as aliasing in the final scene. To address this issue, the recent variant mip-NeRF proposes an Integrated Positional Encoding (IPE) based on a conical view frustum. Although this is expressed with an integral formulation, mip-NeRF instead approximates this integral as the expected value of a multivariate Gaussian distribution. This approximation is reliable for short frustums but degrades with highly elongated regions, which arises when dealing with distant scene objects under a larger depth of field. In this paper, we explore the use of an exact approach for calculating the IPE by using a pyramid-based integral formulation instead of an approximated conical-based one. We denote this formulation as Exact-NeRF and contribute the first approach to offer a precise analytical solution to the IPE within the NeRF domain. Our exploratory work illustrates that such an exact formulation Exact-NeRF matches the accuracy of mip-NeRF and furthermore provides a natural extension to more challenging scenarios without further modification, such as in the case of unbounded scenes. Our contribution aims to both address the hitherto unexplored issues of frustum approximation in earlier NeRF work and additionally provide insight into the potential future consideration of analytical solutions in future NeRF extensions.