Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multi-Agent Pose Uncertainty: A Differentiable Rendering Cramér-Rao Bound

Arun Muthukkumar

TL;DR

This work derives a render-aware Fisher Information Matrix and Cramér-Rao Bound for camera pose on $SE(3)$ by linearizing the differentiable image formation $I=R(\theta;x)+\eta$ around a perturbation $\xi$. The resulting bound $\mathrm{Cov}(\hat{\xi}) \succeq \mathcal{I}(x)^{-1}$ connects dense neural rendering to classical BA uncertainty and naturally extends to multi-camera setups through information fusion. It demonstrates that pose identifiability depends on texture and geometry, with BA equivalence in the pinhole limit and degeneracies captured by the pseudoinverse when $J$ is rank-deficient. Empirical results show the CRB tracks observed pose error across texture regimes and enable cooperative view planning via tile-level FI aggregation, offering a principled diagnostic and design tool for cooperative perception and novel view synthesis.

Abstract

Pose estimation is essential for many applications within computer vision and robotics. Despite its uses, few works provide rigorous uncertainty quantification for poses under dense or learned models. We derive a closed-form lower bound on the covariance of camera pose estimates by treating a differentiable renderer as a measurement function. Linearizing image formation with respect to a small pose perturbation on the manifold yields a render-aware Cramér-Rao bound. Our approach reduces to classical bundle-adjustment uncertainty, ensuring continuity with vision theory. It also naturally extends to multi-agent settings by fusing Fisher information across cameras. Our statistical formulation has downstream applications for tasks such as cooperative perception and novel view synthesis without requiring explicit keypoint correspondences.

Multi-Agent Pose Uncertainty: A Differentiable Rendering Cramér-Rao Bound

TL;DR

This work derives a render-aware Fisher Information Matrix and Cramér-Rao Bound for camera pose on by linearizing the differentiable image formation around a perturbation . The resulting bound connects dense neural rendering to classical BA uncertainty and naturally extends to multi-camera setups through information fusion. It demonstrates that pose identifiability depends on texture and geometry, with BA equivalence in the pinhole limit and degeneracies captured by the pseudoinverse when is rank-deficient. Empirical results show the CRB tracks observed pose error across texture regimes and enable cooperative view planning via tile-level FI aggregation, offering a principled diagnostic and design tool for cooperative perception and novel view synthesis.

Abstract

Pose estimation is essential for many applications within computer vision and robotics. Despite its uses, few works provide rigorous uncertainty quantification for poses under dense or learned models. We derive a closed-form lower bound on the covariance of camera pose estimates by treating a differentiable renderer as a measurement function. Linearizing image formation with respect to a small pose perturbation on the manifold yields a render-aware Cramér-Rao bound. Our approach reduces to classical bundle-adjustment uncertainty, ensuring continuity with vision theory. It also naturally extends to multi-agent settings by fusing Fisher information across cameras. Our statistical formulation has downstream applications for tasks such as cooperative perception and novel view synthesis without requiring explicit keypoint correspondences.
Paper Structure (20 sections, 4 theorems, 7 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 20 sections, 4 theorems, 7 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Differentiable Rendering for Pose Estimation.
  4. Uncertainty Quantification in Neural Rendering.
  5. Information-Theoretic Analyses of Camera Pose.
  6. Multi-Agent and Cooperative Perception.
  7. Manifold and Statistical Estimation Foundations.
  8. Methodology
  9. Core Derivation
  10. Reparameterization invariance.
  11. Identifiability.
  12. Classical BA as a special case.
  13. Multi-Agent Extension
  14. Computing $J$ in practice (autodiff and VJPs)
  15. Complexity and scalability.
  16. ...and 5 more sections

Key Result

Theorem 1

Under the Gaussian model eq:obs and linearization eq:lin, the Fisher Information Matrix (FIM) for the local pose parameter $\xi$ is and the (unbiased) Cramér--Rao bound (CRB) on the local pose covariance is If $\mathcal{I}(x)$ is singular, interpret eq:crb using the Moore--Penrose pseudoinverse $\mathcal{I}(x)^+$.

Figures (3)

  • Figure 1: Pipeline: fixed scene $\theta$ and pose $x$$\rightarrow$ render $I$; autodiff gives $J=\partial R/\partial\xi$; FIM $J^\top\Sigma^{-1}J$; pose CRB $I(x)^{-1}$; interpret as ellipsoids in rotation/translation.
  • Figure 2: A) Multi-agent fusion of Fisher information. B) Adjoint transport from local to global tangent. C) Bandwidth-aware tile selection under budget constraints.
  • Figure 3: CRB calibration and cooperative gains. Left: Coverage vs. nominal confidence shows calibration in high-texture scenes and under-coverage in low-texture ones. Right: log-det information grows submodularly with budget; greedy selection outperforms random and per-agent baselines.

Theorems & Definitions (4)

  • Theorem 1: Render-aware Fisher information on $\mathrm{SE}(3)$
  • Proposition 1: Invariance to smooth minimal pose parametrization
  • Lemma 1: Local identifiability and degeneracy
  • Corollary 1: Bundle adjustment (BA) limit