On the Relation between two Rotation Metrics
This work addresses a foundational aspect for a class of global optimization algorithms in computer vision, but it is incremental as it focuses on proving an existing lemma.
The paper tackles the problem of proving the relationship between two rotation metrics used in global optimization for geometric computer vision, specifically Lemma 2 from Hartley and Kahl's work, and provides a proof based on Rodrigues' Rotation Theorem.
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.