On the Tightness of Semidefinite Relaxations for Rotation Estimation

TL;DR

This paper investigates why semidefinite relaxations are highly effective for rotation estimation problems in computer vision and robotics, providing a theoretical framework and identifying conditions for tightness and failure cases.

Contribution

It introduces an algebraic geometry-based framework to analyze the tightness of semidefinite relaxations for rotation problems, revealing when they succeed or fail.

Findings

Semidefinite relaxations are generally successful but can fail in certain cases.

Some problem classes are always tight with proper parametrization.

Numerical simulations support the theoretical analysis.

Abstract

Why is it that semidefinite relaxations have been so successful in numerous applications in computer vision and robotics for solving non-convex optimization problems involving rotations? In studying the empirical performance we note that there are few failure cases reported in the literature, in particular for estimation problems with a single rotation, motivating us to gain further theoretical understanding. A general framework based on tools from algebraic geometry is introduced for analyzing the power of semidefinite relaxations of problems with quadratic objective functions and rotational constraints. Applications include registration, hand-eye calibration and rotation averaging. We characterize the extreme points, and show that there exist failure cases for which the relaxation is not tight, even in the case of a single rotation. We also show that some problem classes are always tight given an appropriate parametrization. Our theoretical findings are accompanied with numerical simulations, providing further evidence and understanding of the results.