Current methods based on Neural Radiance Fields (NeRF) significantly lack the capacity to quantify uncertainty in their predictions, particularly on the unseen space including the occluded and outside scene content. This limitation hinders their extensive applications in robotics, where the reliability of model predictions has to be considered for tasks such as robotic exploration and planning in unknown environments. To address this, we propose a novel approach to estimate a 3D Uncertainty Field based on the learned incomplete scene geometry, which explicitly identifies these unseen regions. By considering the accumulated transmittance along each camera ray, our Uncertainty Field infers 2D pixel-wise uncertainty, exhibiting high values for rays directly casting towards occluded or outside the scene content. To quantify the uncertainty on the learned surface, we model a stochastic radiance field. Our experiments demonstrate that our approach is the only one that can explicitly reason about high uncertainty both on 3D unseen regions and its involved 2D rendered pixels, compared with recent methods. Furthermore, we illustrate that our designed uncertainty field is ideally suited for real-world robotics tasks, such as next-best-view selection.

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