On the closed-form solution of the rotation matrix arising in computer vision problems

TL;DR

This paper derives a closed-form solution for estimating the optimal rotation matrix in computer vision, clarifying its historical development, mathematical proof, and conditions for uniqueness, which is crucial for many vision tasks.

Contribution

It provides a comprehensive proof of the closed-form solution for the rotation matrix maximization problem, including degenerate cases and uniqueness conditions.

Findings

Derived the closed-form solution for the rotation matrix problem

Analyzed degenerate cases and conditions for uniqueness

Summarized the historical evolution of the problem

Abstract

We show the closed-form solution to the maximization of trace(A'R), where A is given and R is unknown rotation matrix. This problem occurs in many computer vision tasks involving optimal rotation matrix estimation. The solution has been continuously reinvented in different fields as part of specific problems. We summarize the historical evolution of the problem and present the general proof of the solution. We contribute to the proof by considering the degenerate cases of A and discuss the uniqueness of R.