# Exploring the Adjugate Matrix Approach to Quaternion Pose Extraction

arXiv: https://arxiv.org/abs/2205.09116 · HTML: https://mlsanity.com/paper/2205.09116/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 35 references that are in the map.
- First posted on arXiv 17 May 2022.

## Builds on (most cited first, 8 of 8)

- [On the Continuity of Rotation Representations in Neural Networks](https://mlsanity.com/paper/1812.07035/) (2018)
- [PoseCNN: A Convolutional Neural Network for 6D Object Pose Estimation in Cluttered Scenes](https://mlsanity.com/paper/1711.00199/) (2017)
- [Single-Stage 6D Object Pose Estimation](https://mlsanity.com/paper/1911.08324/) (2019)
- [A Smooth Representation of Belief over SO(3) for Deep Rotation Learning with Uncertainty](https://mlsanity.com/paper/2006.01031/) (2020)
- [MLPnP - A Real-Time Maximum Likelihood Solution to the Perspective-n-Point Problem](https://mlsanity.com/paper/1607.08112/) (2016)
- [Revisiting the Continuity of Rotation Representations in Neural Networks](https://mlsanity.com/paper/2006.06234/) (2020)
- [Bundle Adjustment Revisited](https://mlsanity.com/paper/1912.03858/) (2019)
- [Computational Methods for Single-Particle Cryo-EM](https://mlsanity.com/paper/2003.13828/) (2020)

## Built on by (most cited first, 1 of 1)

- [The Determinant Ratio Matrix Approach to Solving 3D Matching and 2D Orthographic Projection Alignment Tasks](https://mlsanity.com/paper/2511.19511/) (2025)

## Abstract (arXiv)

Quaternions are important for a wide variety of rotation-related problems in computer graphics, machine vision, and robotics. We study the nontrivial geometry of the relationship between quaternions and rotation matrices by exploiting the adjugate matrix of the characteristic equation of a related eigenvalue problem to obtain the manifold of the space of a quaternion eigenvector. We argue that quaternions parameterized by their corresponding rotation matrices cannot be expressed, for example, in machine learning tasks, as single-valued functions: the quaternion solution must instead be treated as a manifold, with different algebraic solutions for each of several single-valued sectors represented by the adjugate matrix. We conclude with novel constructions exploiting the quaternion adjugate variables to revisit several classic pose estimation applications: 2D point-cloud matching, 2D point-cloud-to-projection matching, 3D point-cloud matching, 3D orthographic point-cloud-to-projection matching, and 3D perspective point-cloud-to-projection matching. We find an exact solution to the 3D orthographic least squares pose extraction problem, and apply it successfully also to the perspective pose extraction problem with results that improve on existing methods.