Probabilistic Parameter Estimators and Calibration Metrics for Pose Estimation from Image Features

TL;DR

This work develops probabilistic estimators for pose estimation under measurement uncertainty in real-time vision systems, applying them to autonomous runway approaches. It introduces three estimators—LSQ sampling, linear approximation, and MCMC—along with closed-form calibration and sharpness metrics for multivariate normal predictions, and demonstrates Kalman-filter integration for sequential pose tracking. The Linear Approximation estimator is fastest and performs well under Gaussian noise but can be overconfident in some scenarios, while the MCMC approach provides robust, calibrated, and sharp estimates under non-Gaussian noise. The results indicate that probabilistic pose estimates can meaningfully improve sharpness in a Kalman-filtered runway approach, supporting rigorous uncertainty-aware sensor integration in safety-critical aircraft systems and informing certification pathways.

Abstract

This paper addresses the challenge of probabilistic parameter estimation given measurement uncertainty in real-time. We provide a general formulation and apply this to pose estimation for an autonomous visual landing system. We present three probabilistic parameter estimators: a least-squares sampling approach, a linear approximation method, and a probabilistic programming estimator. To evaluate these estimators, we introduce novel closed-form expressions for measuring calibration and sharpness specifically for multivariate normal distributions. Our experimental study compares the three estimators under various noise conditions. We demonstrate that the linear approximation estimator can produce sharp and well-calibrated pose predictions significantly faster than the other methods but may yield overconfident predictions in certain scenarios. Additionally, we demonstrate that these estimators can be integrated with a Kalman filter for continuous pose estimation during a runway approach where we observe a 50\% improvement in sharpness while maintaining marginal calibration. This work contributes to the integration of data-driven computer vision models into complex safety-critical aircraft systems and provides a foundation for developing rigorous certification guidelines for such systems.

Figures (4)

• Figure 1: Known world points $\tl_map_inline:nn{x} { \tl_if_in:VnTF ABCDEFGHIJKLMNOPQRSTUVWXYZ abcdefghijklmnopqrstuvwxyz 0123456789 {x} { \mathbf{x} } { \tl_if_in:VnTF \Gamma\Delta\Theta\Lambda\Pi\Sigma\Upsilon\Phi\Chi\Psi\Omega {x} { \bm{x} } { \tl_if_in:VnTF \alpha\beta\gamma\delta\epsilon\zeta\eta\theta\iota\kappa \lambda\mu\nu\xi\pi\rho\sigma\tau\upsilon\phi\chi\psi\omega \varepsilon\vartheta\varpi\varphi\varsigma\varrho {x} { \bm{\use:c{up}} } { \tl_if_in:VnTF \ell {x} { \boldsymbol{x} } { x } } } } } _{i}$ are projected onto a camera at $\tl_map_inline:nn{\beta} { \tl_if_in:VnTF ABCDEFGHIJKLMNOPQRSTUVWXYZ abcdefghijklmnopqrstuvwxyz 0123456789 {\beta} { \mathbf{\beta} } { \tl_if_in:VnTF \Gamma\Delta\Theta\Lambda\Pi\Sigma\Upsilon\Phi\Chi\Psi\Omega {\beta} { \bm{\beta} } { \tl_if_in:VnTF \alpha\beta\gamma\delta\epsilon\zeta\eta\theta\iota\kappa \lambda\mu\nu\xi\pi\rho\sigma\tau\upsilon\phi\chi\psi\omega \varepsilon\vartheta\varpi\varphi\varsigma\varrho {\beta} { \bm{\use:c{upbeta}} } { \tl_if_in:VnTF \ell {\beta} { \boldsymbol{\beta} } { \beta } } } } }$, where they are measured as $\tl_map_inline:nn{y} { \tl_if_in:VnTF ABCDEFGHIJKLMNOPQRSTUVWXYZ abcdefghijklmnopqrstuvwxyz 0123456789 {y} { \mathbf{y} } { \tl_if_in:VnTF \Gamma\Delta\Theta\Lambda\Pi\Sigma\Upsilon\Phi\Chi\Psi\Omega {y} { \bm{y} } { \tl_if_in:VnTF \alpha\beta\gamma\delta\epsilon\zeta\eta\theta\iota\kappa \lambda\mu\nu\xi\pi\rho\sigma\tau\upsilon\phi\chi\psi\omega \varepsilon\vartheta\varpi\varphi\varsigma\varrho {y} { \bm{\use:c{up}} } { \tl_if_in:VnTF \ell {y} { \boldsymbol{y} } { y } } } } } _{i}$ under the influence of noise. We wish to determine $\tl_map_inline:nn{\beta} { \tl_if_in:VnTF ABCDEFGHIJKLMNOPQRSTUVWXYZ abcdefghijklmnopqrstuvwxyz 0123456789 {\beta} { \mathbf{\beta} } { \tl_if_in:VnTF \Gamma\Delta\Theta\Lambda\Pi\Sigma\Upsilon\Phi\Chi\Psi\Omega {\beta} { \bm{\beta} } { \tl_if_in:VnTF \alpha\beta\gamma\delta\epsilon\zeta\eta\theta\iota\kappa \lambda\mu\nu\xi\pi\rho\sigma\tau\upsilon\phi\chi\psi\omega \varepsilon\vartheta\varpi\varphi\varsigma\varrho {\beta} { \bm{\use:c{upbeta}} } { \tl_if_in:VnTF \ell {\beta} { \boldsymbol{\beta} } { \beta } } } } }$ or a distribution over $\tl_map_inline:nn{\beta} { \tl_if_in:VnTF ABCDEFGHIJKLMNOPQRSTUVWXYZ abcdefghijklmnopqrstuvwxyz 0123456789 {\beta} { \mathbf{\beta} } { \tl_if_in:VnTF \Gamma\Delta\Theta\Lambda\Pi\Sigma\Upsilon\Phi\Chi\Psi\Omega {\beta} { \bm{\beta} } { \tl_if_in:VnTF \alpha\beta\gamma\delta\epsilon\zeta\eta\theta\iota\kappa \lambda\mu\nu\xi\pi\rho\sigma\tau\upsilon\phi\chi\psi\omega \varepsilon\vartheta\varpi\varphi\varsigma\varrho {\beta} { \bm{\use:c{upbeta}} } { \tl_if_in:VnTF \ell {\beta} { \boldsymbol{\beta} } { \beta } } } } }$.
• Figure 2: Uncorrelated normal noise: Calibration and sharpness results.
• Figure 3: Long tail noise: Calibration and sharpness results.
• Figure 4: Kalman filter integration: Calibration results for filtered marginal and joint estimates, and sharpness results for both unfiltered (dotted) and filtered (solid) estimates along a single approach.