On the Relation between two Rotation Metrics

TL;DR

This paper proves a key relationship between rotation metrics used in global optimization algorithms in computer vision, based on Rodrigues' Rotation Theorem, clarifying foundational aspects of rotation space search methods.

Contribution

It provides a formal proof of the relationship between rotation metrics in matrix and axis-angle representations, underpinning optimization algorithms in computer vision.

Findings

Proof of the relationship between rotation metrics

Clarification of Lemma 2 in Hartley and Kahl's work

Foundation for rotation space search algorithms

Abstract

In their work "Global Optimization through Rotation Space Search", Richard Hartley and Fredrik Kahl introduce a global optimization strategy for problems in geometric computer vision, based on rotation space search using a branch-and-bound algorithm. In its core, Lemma 2 of their publication is the important foundation for a class of global optimization algorithms, which is adopted over a wide range of problems in subsequent publications. This lemma relates a metric on rotations represented by rotation matrices with a metric on rotations in axis-angle representation. This work focuses on a proof for this relationship, which is based on Rodrigues' Rotation Theorem for the composition of rotations in axis-angle representation.