On the Tightness of Semidefinite Relaxations for Rotation Estimation
Artikel i vetenskaplig tidskrift, 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


Rotation Estimation

SDP Relaxations



Lucas Brynte

Digitala bildsystem och bildanalys

Viktor Larsson

Eidgenössische Technische Hochschule Zürich (ETH)

José Pedro Lopes Iglesias

Digitala bildsystem och bildanalys

Carl Olsson

Digitala bildsystem och bildanalys

Lunds universitet

Fredrik Kahl

Digitala bildsystem och bildanalys

Journal of Mathematical Imaging and Vision

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

Vol. 64 1 57-67

Optimeringsmetoder med prestandagarantier för maskininlärningsmetoder

Vetenskapsrådet (VR) (2018-05375), 2019-01-01 -- 2022-12-31.


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Robotteknik och automation



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