# Closed-Form Optimal Two-View Triangulation Based on Angular Errors

## TL;DR

This paper introduces the first closed-form, globally optimal solutions for two-view triangulation based on angular errors, applicable to various camera types and requiring less computation.

## Contribution

It presents novel closed-form solutions for optimal two-view triangulation using angular errors, a first in the field, applicable to any central camera model.

## Key findings

- Solutions guarantee global optimality under $L_1$ and $L_
Infinity$ norms.
- Applicable to perspective, fisheye, and omnidirectional cameras.
- Experimental validation confirms theoretical accuracy and efficiency.

## Abstract

In this paper, we study closed-form optimal solutions to two-view triangulation with known internal calibration and pose. By formulating the triangulation problem as $L_1$ and $L_\infty$ minimization of angular reprojection errors, we derive the exact closed-form solutions that guarantee global optimality under respective cost functions. To the best of our knowledge, we are the first to present such solutions. Since the angular error is rotationally invariant, our solutions can be applied for any type of central cameras, be it perspective, fisheye or omnidirectional. Our methods also require significantly less computation than the existing optimal methods. Experimental results on synthetic and real datasets validate our theoretical derivations.