# Reducing Shape-Radiance Ambiguity in Radiance Fields with a Closed-Form Color Estimation Method

arXiv: https://arxiv.org/abs/2312.12726 · HTML: https://mlsanity.com/paper/2312.12726/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 33 references that are in the map.
- First posted on arXiv 20 Dec 2023.

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

- [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)
- [Plenoxels: Radiance Fields without Neural Networks](https://mlsanity.com/paper/2112.05131/) (2021)
- [TensoRF: Tensorial Radiance Fields](https://mlsanity.com/paper/2203.09517/) (2022)
- [NeRF in the Wild: Neural Radiance Fields for Unconstrained Photo Collections](https://mlsanity.com/paper/2008.02268/) (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)
- [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)
- [Depth-supervised NeRF: Fewer Views and Faster Training for Free](https://mlsanity.com/paper/2107.02791/) (2021)
- [KiloNeRF: Speeding up Neural Radiance Fields with Thousands of Tiny MLPs](https://mlsanity.com/paper/2103.13744/) (2021)
- [Local Light Field Fusion: Practical View Synthesis with Prescriptive Sampling Guidelines](https://mlsanity.com/paper/1905.00889/) (2019)
- [Ref-NeRF: Structured View-Dependent Appearance for Neural Radiance Fields](https://mlsanity.com/paper/2112.03907/) (2021)
- [FastNeRF: High-Fidelity Neural Rendering at 200FPS](https://mlsanity.com/paper/2103.10380/) (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)
- [MobileNeRF: Exploiting the Polygon Rasterization Pipeline for Efficient Neural Field Rendering on Mobile Architectures](https://mlsanity.com/paper/2208.00277/) (2022)
- [Dense Depth Priors for Neural Radiance Fields from Sparse Input Views](https://mlsanity.com/paper/2112.03288/) (2021)
- [InfoNeRF: Ray Entropy Minimization for Few-Shot Neural Volume Rendering](https://mlsanity.com/paper/2112.15399/) (2021)
- [Stable View Synthesis](https://mlsanity.com/paper/2011.07233/) (2020)
- [Mixed Neural Voxels for Fast Multi-view Video Synthesis](https://mlsanity.com/paper/2212.00190/) (2022)
- [EfficientNeRF: Efficient Neural Radiance Fields](https://mlsanity.com/paper/2206.00878/) (2022)
- [Improved Direct Voxel Grid Optimization for Radiance Fields Reconstruction](https://mlsanity.com/paper/2206.05085/) (2022)
- [SCADE: NeRFs from Space Carving with Ambiguity-Aware Depth Estimates](https://mlsanity.com/paper/2303.13582/) (2023)
- [DINER: Depth-aware Image-based NEural Radiance fields](https://mlsanity.com/paper/2211.16630/) (2022)
- [Evaluate Geometry of Radiance Fields with Low-frequency Color Prior](https://mlsanity.com/paper/2304.04351/) (2023)

## Built on by (most cited first, 2 of 2)

- [When 3D Gaussian Splatting Recovers Real Surfaces](https://mlsanity.com/paper/2608.30054/) (2026)
- [Robust SG-NeRF: Robust Scene Graph Aided Neural Surface Reconstruction](https://mlsanity.com/paper/2411.13620/) (2024)

## Abstract (arXiv)

Neural radiance field (NeRF) enables the synthesis of cutting-edge realistic novel view images of a 3D scene. It includes density and color fields to model the shape and radiance of a scene, respectively. Supervised by the photometric loss in an end-to-end training manner, NeRF inherently suffers from the shape-radiance ambiguity problem, i.e., it can perfectly fit training views but does not guarantee decoupling the two fields correctly. To deal with this issue, existing works have incorporated prior knowledge to provide an independent supervision signal for the density field, including total variation loss, sparsity loss, distortion loss, etc. These losses are based on general assumptions about the density field, e.g., it should be smooth, sparse, or compact, which are not adaptive to a specific scene. In this paper, we propose a more adaptive method to reduce the shape-radiance ambiguity. The key is a rendering method that is only based on the density field. Specifically, we first estimate the color field based on the density field and posed images in a closed form. Then NeRF's rendering process can proceed. We address the problems in estimating the color field, including occlusion and non-uniformly distributed views. Afterward, it is applied to regularize NeRF's density field. As our regularization is guided by photometric loss, it is more adaptive compared to existing ones. Experimental results show that our method improves the density field of NeRF both qualitatively and quantitatively. Our code is available at https://github.com/qihangGH/Closed-form-color-field.