On the Tightness of Semidefinite Relaxations for Rotation Estimation
Journal article, 2022

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.

Almost Minimal Varieties

Algebraic Geometry

Sum-of-Squares

Rotation Estimation

SDP Relaxations

Duality

Author

Lucas Brynte

Imaging and Image Analysis

Viktor Larsson

Swiss Federal Institute of Technology in Zürich (ETH)

José Pedro Lopes Iglesias

Imaging and Image Analysis

Carl Olsson

Imaging and Image Analysis

Lund University

Fredrik Kahl

Imaging and Image Analysis

Journal of Mathematical Imaging and Vision

0924-9907 (ISSN) 1573-7683 (eISSN)

Vol. 64 1 57-67

Optimization Methods with Performance Guarantees for Subspace Learning

Swedish Research Council (VR) (2018-05375), 2019-01-01 -- 2022-12-31.

Areas of Advance

Information and Communication Technology

Subject Categories

Computational Mathematics

Other Mathematics

Robotics

DOI

10.1007/s10851-021-01054-y

More information

Latest update

3/23/2022