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.
Highlights
Optimization over the special orthogonal group of the orthogonal matrices with determinant one occurs in many geometric vision problems where rigidity of a model needs to be preserved under transformations
We focus on the converse question: For what problem classes can we find objective functions that give a nonzero duality gap and a non-tight relaxation? We use tools from algebraic geometry for analyzing when this happens in the case of general quadratic objective functions over rotational constraints
– We present a novel analysis of the duality properties for quadratic minimization problems subject to rotational constraints based on algebraic geometry
Summary
Optimization over the special orthogonal group of the orthogonal matrices with determinant one occurs in many geometric vision problems where rigidity of a model needs to be preserved under transformations. Recent studies [9,12,14,18,28,32] have observed that in many practical applications the lower bound provided by the dual problem is often the same as the optimal value of the primal one. In such cases, duality offers a way of solving the original non-convex problem using a tight convex. We consider the dual of the dual wherein all quadratic terms of the primal problem are replaced by linear terms over a set of ‘lifted’ variables subject to semidefinite constraints. – We show that the registration problem and the hand– eye calibration problem with SO(2)-parametrization or quaternion parametrization are always guaranteed to produce tight SDP-relaxations
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