旋转估计半定松弛的紧性
On the Tightness of Semidefinite Relaxations for Rotation Estimation
摘要：
为什么半定松弛在计算机视觉和机器人学中的许多应用中如此成功，用于解决涉及旋转的非凸优化问题？在研究的经验表现，我们注意到，有几个失败的情况下，在文献中报道，特别是一个单一的旋转估计问题，激励我们获得进一步的理论理解。基于代数几何工具，提出了一个分析二次目标函数和旋转约束问题半定松弛幂的一般框架。应用包括注册，手眼校准和旋转平均。我们的特点的极端点，并表明，存在故障的情况下，放松是不紧，即使在一个单一的旋转的情况下。我们还表明，一些问题类总是紧给定一个适当的参数化。我们的理论研究结果伴随着数值模拟，提供进一步的证据和理解的结果。
英文摘要：
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.
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