Branch-and-bound-based consensus maximization stands out due to its important ability of retrieving the globally optimal solution to outlier-affected geometric problems. However, while the discovery of such solutions caries high scientific value, its application in practical scenarios is often prohibited by its computational complexity growing exponentially as a function of the dimensionality of the problem at hand. In this work, we convey a novel, general technique that allows us to branch over an $n-1$ dimensional space for an n-dimensional problem. The remaining degree of freedom can be solved globally optimally within each bound calculation by applying the efficient interval stabbing technique. While each individual bound derivation is harder to compute owing to the additional need for solving a sorting problem, the reduced number of intervals and tighter bounds in practice lead to a significant reduction in the overall number of required iterations. Besides an abstract introduction of the approach, we present applications to three fundamental geometric computer vision problems: camera resectioning, relative camera pose estimation, and point set registration. Through our exhaustive tests, we demonstrate significant speed-up factors at times exceeding two orders of magnitude, thereby increasing the viability of globally optimal consensus maximizers in online application scenarios.

翻译：摘要：基于分支定界的共识最大化方法因其能够为受离群点影响的几何问题检索全局最优解而具有重要价值。尽管此类解的发现具有极高科学价值，但其在实际场景中的应用常受限于计算复杂度——该复杂度随问题维度呈指数增长。本文提出一种新颖的通用技术，允许针对n维问题在n-1维空间中进行分支操作。通过应用高效的区间刺穿技术，剩余的一个自由度可在每次边界计算中以全局最优方式求解。尽管由于需额外解决排序问题，单次边界推导的计算难度增加，但实际应用中区间数量的减少与更紧致的边界显著降低了整体迭代次数。除对该方法的抽象介绍外，我们还将其应用于三个基础几何计算机视觉问题：相机位姿标定、相对相机位姿估计和点集配准。通过全面测试，我们证明了该方法能实现高达两个数量级以上的加速比例，从而显著提升了全局最优共识最大化算法在在线场景中的可行性。