LOSTU: Fast, Scalable, and Uncertainty-Aware Triangulation

TL;DR

LOSTU addresses robust, scalable triangulation under camera pose uncertainties by introducing a linear, non-iterative maximum-likelihood framework that unifies with DLT and prior LOST variants. It extends the Linear Optimal Sine framework to incorporate camera-parameter uncertainties through a weighted residual, enabling principled covariance propagation and closed-form solutions. Across two-view and multi-view experiments, LOSTU often matches or exceeds Levenberg-Marquardt refinements in accuracy while dramatically reducing computation time, especially when covariances are available. In sequential SfM and SLAM contexts (e.g., ETH3D, Vesta), properly propagated covariances with LOSTU improve point counts and fidelity, underscoring the practical impact of uncertainty-aware triangulation on large-scale reconstruction tasks.

Abstract

This work proposes a non-iterative, scalable, and statistically optimal way to triangulate called \texttt{LOSTU}. Unlike triangulation algorithms that minimize the reprojection ($L_2$) error, LOSTU will still provide the maximum likelihood estimate when there are errors in camera pose or parameters. This generic framework is used to contextualize other triangulation methods like the direct linear transform (DLT) or the midpoint. Synthetic experiments show that LOSTU can be substantially faster than using uncertainty-aware Levenberg-Marquardt (or similar) optimization schemes, while providing results of comparable precision. Finally, LOSTU is implemented in sequential reconstruction in conjunction with uncertainty-aware pose estimation, where it yields better reconstruction metrics.

Figures (4)

• Figure 1: Percentage of RMSE deterioration with respect to HS for two view triangulation. Negative value means better than minimizing the reprojection error. Curves for DLT and midpoint overlap. We observe a clear gap between optimal and suboptimal methods.
• Figure 2: Percentage of RMSE deterioration with respect to DLT+LM (reproj) for 50-view triangulation. Negative value means better than minimizing the reprojection error.
• Figure 3: Evolution of triangulation as the number of views increases. (a): All techniques exhibit linear runtime complexity but have different slopes. (b) Percentage of RMSE deterioration with respect to DLT+LM (reproj). Legends can be found in \ref{['fig:n-view']}
• Figure 4: EPnPU+LOSTU reconstruction for facade (left) and delivery_area (right) schoeps:2017_ETH3D.