# Stochastic Neural Radiance Fields: Quantifying Uncertainty in Implicit 3D Representations

arXiv: https://arxiv.org/abs/2109.02123 · HTML: https://mlsanity.com/paper/2109.02123/

## Key facts

- Cited by 82 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 40 references that are in the map.
- First posted on arXiv 5 Sep 2021.

## Builds on (most cited first, 30 of 31)

- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [pixelNeRF: Neural Radiance Fields from One or Few Images](https://mlsanity.com/paper/2012.02190/) (2020)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [D-NeRF: Neural Radiance Fields for Dynamic Scenes](https://mlsanity.com/paper/2011.13961/) (2020)
- [NeRF in the Wild: Neural Radiance Fields for Unconstrained Photo Collections](https://mlsanity.com/paper/2008.02268/) (2020)
- [Neural Sparse Voxel Fields](https://mlsanity.com/paper/2007.11571/) (2020)
- [Scene Representation Networks: Continuous 3D-Structure-Aware Neural Scene Representations](https://mlsanity.com/paper/1906.01618/) (2019)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [GIRAFFE: Representing Scenes as Compositional Generative Neural Feature Fields](https://mlsanity.com/paper/2011.12100/) (2020)
- [Neural Scene Flow Fields for Space-Time View Synthesis of Dynamic Scenes](https://mlsanity.com/paper/2011.13084/) (2020)
- [Neural Body: Implicit Neural Representations with Structured Latent Codes for Novel View Synthesis of Dynamic Humans](https://mlsanity.com/paper/2012.15838/) (2020)
- [A Simple Baseline for Bayesian Uncertainty in Deep Learning](https://mlsanity.com/paper/1902.02476/) (2019)
- [Variational Dropout and the Local Reparameterization Trick](https://mlsanity.com/paper/1506.02557/) (2015)
- [Dynamic Neural Radiance Fields for Monocular 4D Facial Avatar Reconstruction](https://mlsanity.com/paper/2012.03065/) (2020)
- [Space-time Neural Irradiance Fields for Free-Viewpoint Video](https://mlsanity.com/paper/2011.12950/) (2020)
- [INeRF: Inverting Neural Radiance Fields for Pose Estimation](https://mlsanity.com/paper/2012.05877/) (2020)
- [Probabilistic Backpropagation for Scalable Learning of Bayesian Neural Networks](https://mlsanity.com/paper/1502.05336/) (2015)
- [Mixture of Volumetric Primitives for Efficient Neural Rendering](https://mlsanity.com/paper/2103.01954/) (2021)
- [DONeRF: Towards Real-Time Rendering of Compact Neural Radiance Fields using Depth Oracle Networks](https://mlsanity.com/paper/2103.03231/) (2021)
- [AutoInt: Automatic Integration for Fast Neural Volume Rendering](https://mlsanity.com/paper/2012.01714/) (2020)
- [Posterior Network: Uncertainty Estimation without OOD Samples via Density-Based Pseudo-Counts](https://mlsanity.com/paper/2006.09239/) (2020)
- [NeRF-VAE: A Geometry Aware 3D Scene Generative Model](https://mlsanity.com/paper/2104.00587/) (2021)
- [Bayesian Layers: A Module for Neural Network Uncertainty](https://mlsanity.com/paper/1812.03973/) (2018)
- [PVA: Pixel-aligned Volumetric Avatars](https://mlsanity.com/paper/2101.02697/) (2021)
- [Maximizing Overall Diversity for Improved Uncertainty Estimates in Deep Ensembles](https://mlsanity.com/paper/1906.07380/) (2019)

## Built on by (most cited first, 30 of 82)

- [RegNeRF: Regularizing Neural Radiance Fields for View Synthesis from Sparse Inputs](https://mlsanity.com/paper/2112.00724/) (2021)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [ActiveNeRF: Learning where to See with Uncertainty Estimation](https://mlsanity.com/paper/2209.08546/) (2022)
- [PUP 3D-GS: Principled Uncertainty Pruning for 3D Gaussian Splatting](https://mlsanity.com/paper/2406.10219/) (2024)
- [Bayes' Rays: Uncertainty Quantification for Neural Radiance Fields](https://mlsanity.com/paper/2309.03185/) (2023)
- [Conditional-Flow NeRF: Accurate 3D Modelling with Reliable Uncertainty Quantification](https://mlsanity.com/paper/2203.10192/) (2022)
- [FisherRF: Active View Selection and Uncertainty Quantification for Radiance Fields using Fisher Information](https://mlsanity.com/paper/2311.17874/) (2023)
- [Density-aware NeRF Ensembles: Quantifying Predictive Uncertainty in Neural Radiance Fields](https://mlsanity.com/paper/2209.08718/) (2022)
- [Uncertainty Guided Policy for Active Robotic 3D Reconstruction using Neural Radiance Fields](https://mlsanity.com/paper/2209.08409/) (2022)
- [PhyRecon: Physically Plausible Neural Scene Reconstruction](https://mlsanity.com/paper/2404.16666/) (2024)
- [Active Neural Mapping](https://mlsanity.com/paper/2308.16246/) (2023)
- [FlipNeRF: Flipped Reflection Rays for Few-shot Novel View Synthesis](https://mlsanity.com/paper/2306.17723/) (2023)
- [4D Gaussian Splatting in the Wild with Uncertainty-Aware Regularization](https://mlsanity.com/paper/2411.08879/) (2024)
- [Neural Visibility Field for Uncertainty-Driven Active Mapping](https://mlsanity.com/paper/2406.06948/) (2024)
- [Drone-assisted Road Gaussian Splatting with Cross-view Uncertainty](https://mlsanity.com/paper/2408.15242/) (2024)
- [Modeling uncertainty for Gaussian Splatting](https://mlsanity.com/paper/2403.18476/) (2024)
- [GauSS-MI: Gaussian Splatting Shannon Mutual Information for Active 3D Reconstruction](https://mlsanity.com/paper/2504.21067/) (2025)
- [Estimating 3D Uncertainty Field: Quantifying Uncertainty for Neural Radiance Fields](https://mlsanity.com/paper/2311.01815/) (2023)
- [NeRF Director: Revisiting View Selection in Neural Volume Rendering](https://mlsanity.com/paper/2406.08839/) (2024)
- [Leveraging Neural Radiance Fields for Uncertainty-Aware Visual Localization](https://mlsanity.com/paper/2310.06984/) (2023)
- [UP-NeRF: Unconstrained Pose-Prior-Free Neural Radiance Fields](https://mlsanity.com/paper/2311.03784/) (2023)
- [Neural Radiance Field-based Visual Rendering: A Comprehensive Review](https://mlsanity.com/paper/2404.00714/) (2024)
- [Neural Radiance Fields for the Real World: A Survey](https://mlsanity.com/paper/2501.13104/) (2025)
- [The Role of Predictive Uncertainty and Diversity in Embodied AI and Robot Learning](https://mlsanity.com/paper/2405.03164/) (2024)
- [Beyond RGB: Scene-Property Synthesis with Neural Radiance Fields](https://mlsanity.com/paper/2206.04669/) (2022)
- [Sources of Uncertainty in 3D Scene Reconstruction](https://mlsanity.com/paper/2409.06407/) (2024)
- [Bayesian NeRF: Quantifying Uncertainty with Volume Density for Neural Implicit Fields](https://mlsanity.com/paper/2404.06727/) (2024)
- [Multimodal LLM Guided Exploration and Active Mapping using Fisher Information](https://mlsanity.com/paper/2410.17422/) (2024)
- [UC-NeRF: Uncertainty-aware Conditional Neural Radiance Fields from Endoscopic Sparse Views](https://mlsanity.com/paper/2409.02917/) (2024)
- [Learning with Unreliability: Fast Few-shot Voxel Radiance Fields with Relative Geometric Consistency](https://mlsanity.com/paper/2403.17638/) (2024)

## Abstract (arXiv)

Neural Radiance Fields (NeRF) has become a popular framework for learning implicit 3D representations and addressing different tasks such as novel-view synthesis or depth-map estimation. However, in downstream applications where decisions need to be made based on automatic predictions, it is critical to leverage the confidence associated with the model estimations. Whereas uncertainty quantification is a long-standing problem in Machine Learning, it has been largely overlooked in the recent NeRF literature. In this context, we propose Stochastic Neural Radiance Fields (S-NeRF), a generalization of standard NeRF that learns a probability distribution over all the possible radiance fields modeling the scene. This distribution allows to quantify the uncertainty associated with the scene information provided by the model. S-NeRF optimization is posed as a Bayesian learning problem which is efficiently addressed using the Variational Inference framework. Exhaustive experiments over benchmark datasets demonstrate that S-NeRF is able to provide more reliable predictions and confidence values than generic approaches previously proposed for uncertainty estimation in other domains.