# Absolute Triangulation Algorithms for Space Exploration

arXiv: https://arxiv.org/abs/2205.12197 · HTML: https://mlsanity.com/paper/2205.12197/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 119 references that are in the map.
- First posted on arXiv 24 May 2022.

## Builds on (most cited first, 1 of 1)

- [TEASER: Fast and Certifiable Point Cloud Registration](https://mlsanity.com/paper/2001.07715/) (2020)

## Built on by (most cited first, 5 of 5)

- [Global Structure-from-Motion Revisited](https://mlsanity.com/paper/2407.20219/) (2024)
- [Optimal DLT-based Solutions for the Perspective-n-Point](https://mlsanity.com/paper/2410.14164/) (2024)
- [An Autonomous Vision-Based Algorithm for Interplanetary Navigation](https://mlsanity.com/paper/2309.09590/) (2023)
- [LOSTU: Fast, Scalable, and Uncertainty-Aware Triangulation](https://mlsanity.com/paper/2311.11171/) (2023)
- [Principal Component Analysis for Lunar Crater Detection](https://mlsanity.com/paper/2605.17125/) (2026)

## Abstract (arXiv)

Images are an important source of information for spacecraft navigation and for three-dimensional reconstruction of observed space objects. Both of these applications take the form of a triangulation problem when the camera has a known attitude and the measurements extracted from the image are line of sight (LOS) directions. This work provides a comprehensive review of the history and theoretical foundations of triangulation. A variety of classical triangulation algorithms are reviewed, including a number of suboptimal linear methods (many LOS measurements) and the optimal method of Hartley and Sturm (only two LOS measurements). It is shown that the optimal many-measurement case may be solved without iteration as a linear system using the new Linear Optimal Sine Triangulation (LOST) method. Both LOST and the polynomial method of Hartley and Sturm provide the same result in the case of only two measurements. The various triangulation algorithms are assessed with a few numerical examples, including planetary terrain relative navigation, angles-only optical navigation at Uranus, 3-D reconstruction of Notre-Dame de Paris, and angles-only relative navigation.