Open access
Date
2022-01Type
- Journal Article
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. Show more
Permanent link
https://doi.org/10.3929/ethz-b-000519255Publication status
publishedExternal links
Journal / series
Journal of mathematical imaging and visionVolume
Pages / Article No.
Publisher
SpringerSubject
SDP relaxations; Duality; Algebraic geometry; Almost minimal varieties; Sum-of-squares; Rotation estimationMore
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