# Triangulation: Why Optimize?

arXiv: https://arxiv.org/abs/1907.11917 · HTML: https://mlsanity.com/paper/1907.11917/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jul 2019.

## Built on by (most cited first, 9 of 9)

- [HARA: A Hierarchical Approach for Robust Rotation Averaging](https://mlsanity.com/paper/2111.08831/) (2021)
- [Depth-Guided Sparse Structure-from-Motion for Movies and TV Shows](https://mlsanity.com/paper/2204.02509/) (2022)
- [Tracking monocular camera pose and deformation for SLAM inside the human body](https://mlsanity.com/paper/2204.08309/) (2022)
- [From Correspondences to Pose: Non-minimal Certifiably Optimal Relative Pose without Disambiguation](https://mlsanity.com/paper/2312.05995/) (2023)
- [3D Trajectory Reconstruction of Moving Points Based on a Monocular Camera](https://mlsanity.com/paper/2502.19689/) (2025)
- [3D Trajectory Reconstruction of Moving Points Based on Asynchronous Cameras](https://mlsanity.com/paper/2506.00541/) (2025)
- [CuSfM: CUDA-Accelerated Structure-from-Motion](https://mlsanity.com/paper/2510.15271/) (2025)
- [LOSTU: Fast, Scalable, and Uncertainty-Aware Triangulation](https://mlsanity.com/paper/2311.11171/) (2023)
- [A Framework for Reducing the Complexity of Geometric Vision Problems and its Application to Two-View Triangulation with Approximation Bounds](https://mlsanity.com/paper/2503.08142/) (2025)

## Abstract (arXiv)

For decades, it has been widely accepted that the gold standard for two-view triangulation is to minimize the cost based on reprojection errors. In this work, we challenge this idea. We propose a novel alternative to the classic midpoint method that leads to significantly lower 2D errors and parallax errors. It provides a numerically stable closed-form solution based solely on a pair of backprojected rays. Since our solution is rotationally invariant, it can also be applied for fisheye and omnidirectional cameras. We show that for small parallax angles, our method outperforms the state-of-the-art in terms of combined 2D, 3D and parallax accuracy, while achieving comparable speed.