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

Jianxiong Shen, Adria Ruiz, Antonio Agudo, Francesc Moreno-Noguer

arXiv:2109.02123v3cs.CV

TL;DR

NeRF-based predictions often lack confidence estimates, despite their use in downstream decisions where uncertainty matters. S-NeRF models a distribution over radiance fields and uses Bayesian learning with Variational Inference. The paper reports better uncertainty-estimation results than prior approaches, without increased computational complexity.

  • Problem

    NeRF approaches do not provide confidence associated with model outputs, although downstream decisions may need to account for prediction uncertainty.

    Source: §1 · p. 1, §2 · p. 2

  • Method

    S-NeRF generalizes NeRF by learning a distribution over possible radiance fields through Bayesian learning optimized with Variational Inference.

    Source: §1 · pp. 1–2

  • Results

    S-NeRF achieves significantly better results than applicable state-of-the-art approaches without increasing computational complexity, and provides uncertainty estimates for tasks including depth-map estimation.

    Source: §5 · p. 8

  • Takeaways & Limitations

    S-NeRF quantifies uncertainty for both rendered views and estimates related to 3D geometry, unlike NeRF-W’s rendered-pixel variance approach.

    Source: §2 · p. 2

    NeRF-W estimates rendered-pixel variance but does not explicitly model radiance-field uncertainty or quantify confidence in the underlying 3D geometry.

    Source: §2 · p. 2

Abstract

from arXiv · show

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.

1. Introduction

NeRF supports novel-view rendering and downstream scene tasks, but its predictions lack confidence estimates needed for decisions in applications such as robotics. S-NeRF models a distribution over radiance fields and uses Bayesian learning with Variational Inference to estimate uncertainty.

1. Introduction

  • NeRF-based scene representations support tasks including camera-pose recovery [38], semantic segmentation [16], and depth estimation [28], but do not provide confidence associated with their outputs.The introduction notes that robotics decisions must consider both predictions and their uncertainty.Source: §1 · p. 1
  • S-NeRF generalizes NeRF by modeling spatial-location radiance-density pairs as stochastic variables, inducing a distribution over possible radiance fields.The distribution is intended to quantify uncertainty in outputs for tasks such as novel-view rendering and depth-map estimation.Source: §1 · pp. 1–2
  • S-NeRF estimates a Bayesian posterior over radiance fields and uses Variational Inference to make optimization tractable.Source: §1 · p. 2
  • Benchmark experiments report more reliable uncertainty estimates than generic approaches previously proposed for uncertainty estimation in other domains.The evaluation covers novel-view synthesis and depth-map estimation.Source: §1 · p. 2

2. Related Work

Prior NeRF extensions address rendering, scene, illumination, or dynamic-scene concerns, while uncertainty in the implicit representation remains a distinct problem. S-NeRF targets that uncertainty, including confidence about 3D geometry, and its single-network design avoids evaluating multiple model instances.

2. Related Work

  • NeRF uses sparse 2D views to learn an implicit 3D scene representation, predicting density and emitted radiance for rendering novel views or estimating geometry.The passage situates NeRF alongside Scene Representation Networks and Neural Volumes [22], and identifies volume rendering techniques as the means of combining its outputs.Source: §2 · p. 2
  • Prior NeRF extensions address different concerns: speeding training or rendering [21] [20] [29] [23], scene priors [17], illumination and transient occluders [25], and dynamic objects [33] [8] [34] [19] [37] [32].Source: §2 · p. 2
  • NeRF-derived scene information can support robotic navigation or manipulation and augmented-reality applications, but prior approaches cannot quantify uncertainty needed to evaluate downstream decision risk.Source: §2 · p. 2
  • NeRF-W [25] estimates rendered-pixel variance for transient objects, but does not explicitly model radiance-field uncertainty or quantify confidence in underlying 3D geometry.The passage also notes that pixel colors arise from a physical process unrelated to model uncertainty, making their use for variance estimation through volume rendering not theoretically founded.Source: §2 · p. 2
  • S-NeRF is presented as the first approach to explicitly model uncertainty in NeRF’s implicit representation, extending uncertainty estimates from rendered views to 3D-geometry estimates.Source: §2 · p. 2
  • Bayesian Neural Networks estimate posterior distributions over network parameters but are typically computationally expensive and require substantial architecture and training-procedure modifications.Dropout-based methods [9] and deep ensembles [14] are alternative strategies described for implicitly learning parameter distributions.Source: §2 · p. 2
  • Dropout-based and ensemble uncertainty methods require multiple model evaluations, limiting their application where NeRF rendering is already computationally expensive.Source: §2 · p. 3
  • Unlike approaches that learn a posterior over model parameters, S-NeRF uses a single network to encode a distribution over possible radiance fields, avoiding multiple model evaluations for uncertainty estimates.Source: §2 · p. 3

3. Stochastic Neural Radiance Fields

S-NeRF generalizes NeRF by learning a distribution over radiance fields and estimating its posterior from training data. Variational inference makes this optimization tractable, and samples from the learned distribution provide uncertainty estimates for rendered views and depth maps.

3.1. From Standard to Stochastic-NeRF

  • S-NeRF models radiance and density as stochastic variables, learning a distribution over possible radiance fields and a Bayesian posterior given training data.This replaces standard NeRF’s deterministic radiance-density values at each spatial location.Source: §3.1 · p. 3

3.2. Learning S-NeRF with Variational Inference

  • Variational inference approximates the intractable posterior with qθ(F), optimizing expected training likelihood alongside a KL penalty to the prior.The objective combines expected data fit under the approximate posterior with its divergence from the prior.Source: §3.2 · p. 3

3.2.1 Modelling the approximate posterior

  • The approximate posterior is fully factorized, with a network predicting radiance and density distribution parameters; radiance uses logistic-normal and density rectified-normal distributions.These choices match radiance’s [0, 1] bounds and density’s nonnegative support.Source: §3.2.1 · p. 4

3.2.2 Computing the log-likelihood

  • For each training ray, S-NeRF samples radiance-density trajectories, renders their pixel colors, and estimates the color likelihood for stochastic-gradient optimization.The reparameterization trick propagates gradients through the generated samples.Source: §3.2.2 · pp. 4–5, Figure 2 · p. 4

3.2.3 Estimating the posterior-prior KL divergence

  • The posterior-prior KL sums divergences across location-view pairs; training samples pairs to estimate it, with prior variances also optimized along observed rays.For other locations, the radiance and density prior variances are fixed at 10.Source: §3.2.3 · p. 5

3.3. Inference and Uncertainty Estimation

  • At inference, means and variances of rendered color samples give pixel predictions and uncertainty; sampled density trajectories similarly yield depth estimates and uncertainty.Depth samples are expected ray-termination depths computed from density trajectories.Source: §3.3 · p. 5

4. Experiments

The experiments evaluate S-NeRF’s uncertainty estimates for novel-view rendering and depth estimation, including benchmark comparisons and analyses of prior choices. Results indicate improved uncertainty estimation and confidence on unobserved scene regions and depth boundaries.

4.1. Experimental setup

  • Experiments use LLFF’s eight scenes with about 20% of views for training, comparing NeRF-W, MC-dropout, and Deep-Ensembles using NLL and MSE–uncertainty correlation.The dropout and ensemble baselines use five samples or models.Source: §4.1 · pp. 5–6, Table 2 · p. 6

4.2. Uncertainty estimation in novel-view synthesis

  • S-NeRF outperformed prior approaches across scenes and metrics, with a 35% average NLL decrease over the previous state of the art.Table 1 reports uncertainty-estimation metrics on the LLFF dataset.Source: Table 1 · p. 6, §4.2 · p. 6
  • Against NeRF-W, S-NeRF improves uncertainty metrics without increasing rendering time, while MC-dropout and five-model ensembles multiply NeRF’s computational complexity by five.The paper notes that using more dropout samples or ensemble models could improve their estimates but would increase rendering time.Source: §4.2 · p. 7
  • In unseen regions, KL regularization pulls the posterior toward a high-variance prior, allowing S-NeRF to assign high uncertainty where other methods remain underconfident.Figure 4 shows this pattern in the unobserved right side of the test view.Source: §4.2 · p. 7, Figure 4 · p. 7

4.3. Analysing the effect of the prior distribution

  • A high-variance KL prior improves NLL and MSE–uncertainty correlation over no KL, while a learned prior for observed locations further improves performance and stability.A high-variance prior on observed locations can hinder negative log-likelihood minimization; the learned variance addresses this issue.Source: Table 3 · p. 8, §4.3 · p. 8

4.4. Uncertainty estimation in depth-map synthesis

  • Depth uncertainty rises at foreground boundaries where depth changes discontinuously and is low for pixels depicting scene areas absent from training views.The paper presents these patterns as confidence information about the scene’s underlying 3D geometry.Source: Figure 5 · p. 8, §4.4 · p. 8

5. Conclusions

The conclusion presents S-NeRF as a probabilistic generalization of NeRF for uncertainty estimation in neural volume rendering. It reports better results without increased computational complexity and identifies depth estimation and combination with other NeRF methods as supported extensions.

5. Conclusions

  • S-NeRF models a distribution over possible scene radiance fields and achieves better uncertainty-estimation results than state-of-the-art approaches without increasing computational complexity.The conclusion also illustrates uncertainty estimation for depth maps and says the formulation can be combined with existing or future NeRF methods.Source: §5 · p. 8

A. Methods

The appendix introduces its material as technical details for the proposed S-NeRF method described in Section 3.

A. Methods

  • Appendix A provides technical details for the proposed S-NeRF method described in Section 3.Source: A. Methods · p. 11

A.1. Distributions

S-NeRF assigns distributions to radiance and density, then uses reparameterized sampling so their parameters can be optimized by backpropagation.

A.1. Distributions

  • Radiance channels use independent logistic-normal distributions to keep values between 0 and 1, while density follows a rectified normal distribution to remain positive.The network outputs the distribution means and standard deviations.Source: §A.1 · p. 11

A.2. Backpropagation through sampling with the reparametrization-trick

  • A standard-normal variable reparameterizes sampled radiance and density, making gradients of their distribution parameters computable during backpropagation.Radiance uses a sigmoid transform and density uses a rectification transform.Source: §A.2 · p. 11

A.3. The posterior-prior KL divergence

The method approximates the KL divergence between learned posteriors and priors over radiance fields by sampling scene locations and directions.

A.3. The posterior-prior KL divergence

  • Randomly sampling one point from each of N evenly spaced bins per scene axis yields N × N × N location-direction samples for tractable KL computation.The spatial bounds are defined from the captured scene images.Source: §A.3 · p. 11
  • At each sampled location-view pair, the network outputs posterior radiance and density parameters, whose distributions are compared with the corresponding priors using KL divergence.Source: §A.3 · p. 11
  • Monte Carlo estimates approximate the density and radiance KL integrations, with rectification keeping density values positive.The density and radiance integrals are estimated over their respective variables.Source: §A.3 · p. 12

A.4. Trapezoidal rule

The appendix describes a trapezoidal approximation for volume rendering and the sampling and optimization steps in S-NeRF's learning procedure.

A.4. Trapezoidal rule

  • The rendering procedure stratifies each ray into N evenly spaced bins, samples locations, and produces K density-radiance pairs per location for integration.Adjacent sampled locations are separated by δi.Source: §A.4 · p. 12
  • Large variances can produce extremely large densities and optimization instabilities with original NeRF alpha compositing, motivating the alternative trapezoidal approximation.The authors report that the alternative was more stable in preliminary experiments.Source: §A.4 · p. 12

B. Pseudo-algorithm

  • Training samples points along camera rays, predicts radiance and density distributions, renders K pixel-color samples, computes log-likelihood and KL terms, then optimizes network and distribution parameters by SGD.The KL term uses uniformly sampled scene coordinates.Source: B. Pseudo-algorithm · p. 13
  • The learning procedure computes the KL term by uniformly sampling scene coordinates and updates θ and the distribution parameters with SGD.Source: B. Pseudo-algorithm · p. 13

C. Additional Qualitative Results

Additional qualitative results show S-NeRF's uncertainty estimates for rendered views and estimated depth maps across evaluated scenes.

C. Additional Qualitative Results

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