[Paper Review] Stochastic Neural Radiance Fields: Quantifying Uncertainty in Implicit 3D Representations

Jianxiong Shen, Adrià Ruiz+2 more|arXiv (Cornell University)|Sep 5, 2021

Human Pose and Action Recognition4 citations

TL;DR

This paper proposes Stochastic Neural Radiance Fields (S-NeRF), a Bayesian extension of NeRF that models uncertainty by learning a probability distribution over radiance fields via variational inference. S-NeRF provides reliable uncertainty estimates for novel-view synthesis and depth-map estimation, outperforming generic uncertainty methods without increasing computational cost.

ABSTRACT

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.

Motivation & Objective

• To address the lack of uncertainty quantification in NeRF-based 3D scene representations, which limits their use in safety-critical applications like robotics and AR.
• To develop a probabilistic generalization of NeRF that models uncertainty over all possible radiance fields, rather than deterministic predictions.
• To enable reliable confidence estimation for downstream tasks such as novel-view rendering and depth-map prediction.
• To integrate uncertainty estimation into NeRF without increasing computational complexity, ensuring practical deployment.

Proposed method

• S-NeRF generalizes standard NeRF by modeling radiance and density as stochastic variables with a parameterized distribution, enabling uncertainty modeling over all possible radiance fields.
• The method formulates NeRF learning as a Bayesian inference problem, estimating the posterior distribution over radiance fields using variational inference.
• A variational posterior is optimized via stochastic gradient descent, minimizing a variational lower bound that includes negative log-likelihood and KL divergence terms.
• A learned prior with optimizable variance is introduced for spatial locations observed in training views, improving stability and performance over fixed high-variance priors.
• The framework supports uncertainty estimation not only for novel-view rendering but also for 3D geometry, such as depth-map prediction.
• The optimization objective combines log-likelihood, KL divergence, and a learned prior term, enabling better generalization and uncertainty calibration.

Experimental results

Research questions

• RQ1 Can a probabilistic generalization of NeRF effectively quantify uncertainty in 3D scene representations?
• RQ2 How does S-NeRF improve uncertainty estimation compared to generic uncertainty methods in other domains?
• RQ3 Does incorporating a learned prior for observed spatial locations enhance uncertainty calibration and model stability?
• RQ4 Can S-NeRF provide meaningful uncertainty estimates for 3D geometry tasks like depth-map estimation?
• RQ5 Is the proposed method scalable and efficient, maintaining computational efficiency while adding uncertainty modeling?

Key findings

• S-NeRF achieves significantly better uncertainty calibration than generic uncertainty estimation methods, as measured by lower negative log-likelihood (NLL) and higher MSE-uncertainty correlation.
• The inclusion of a learned prior with optimizable variance for observed spatial locations improves both performance and training stability, reducing the risk of sub-optimal solutions from fixed high-variance priors.
• S-NeRF assigns high uncertainty to unobserved regions and areas with depth discontinuities, such as object borders, reflecting low model confidence in these regions.
• The model demonstrates reliable uncertainty estimates in depth-map synthesis, correctly identifying low-confidence regions corresponding to occlusions or scene boundaries.
• Experiments on benchmark datasets show that S-NeRF maintains computational efficiency while providing state-of-the-art uncertainty quantification in NeRF-based 3D representations.
• The framework successfully identifies and flags regions with high uncertainty, such as those affected by transient objects or missing training views, enhancing robustness in real-world applications.