# An Efficient Algebraic Solution to the Perspective-Three-Point Problem

## TL;DR

This paper introduces a new algebraic method for solving the perspective-3-point problem, enabling accurate and robust camera pose estimation with lower computational cost by directly deriving attitude and position from geometric constraints.

## Contribution

The paper presents an algebraic approach that directly computes camera attitude and position from three reference points, improving accuracy and efficiency over previous methods.

## Key findings

- Higher numerical accuracy demonstrated through simulations
- Reduced computational cost compared to existing methods
- Robust performance in near-singular configurations

## Abstract

In this work, we present an algebraic solution to the classical perspective-3-point (P3P) problem for determining the position and attitude of a camera from observations of three known reference points. In contrast to previous approaches, we first directly determine the camera's attitude by employing the corresponding geometric constraints to formulate a system of trigonometric equations. This is then efficiently solved, following an algebraic approach, to determine the unknown rotation matrix and subsequently the camera's position. As compared to recent alternatives, our method avoids computing unnecessary (and potentially numerically unstable) intermediate results, and thus achieves higher numerical accuracy and robustness at a lower computational cost. These benefits are validated through extensive Monte-Carlo simulations for both nominal and close-to-singular geometric configurations.