Multiple-view triangulation by _∞ minimisation has become established in computer vision. State-of-the-art _∞ triangulation algorithms exploit the quasiconvexity of the cost function to derive iterative update rules that deliver the global minimum. Such algorithms, however, can be computationally costly for large problem instances that contain many image measurements, e.g., from web-based photo sharing sites or long-term video recordings. In this paper, we prove that _∞ triangulation admits a coreset approximation scheme, which seeks small representative subsets of the input data called coresets. A coreset possesses the special property that the error of the _∞ solution on the coreset is within known bounds from the global minimum. We establish the necessary mathematical underpinnings of the coreset algorithm, specifically, by enacting the stopping criterion of the algorithm and proving that the resulting coreset gives the desired approximation accuracy. On large-scale triangulation problems, our method provides theoretically sound approximate solutions. Iterated until convergence, our coreset algorithm is also guaranteed to reach the true optimum. On practical datasets, we show that our technique can in fact attain the global minimiser much faster than current methods.

中文翻译：

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三角测量的核心集


通过 _∞ 最小化进行的多视图三角测量已经在计算机视觉中建立。最先进的 _∞ 三角剖分算法利用成本函数的拟凸性来导出提供全局最小值的迭代更新规则。然而，对于包含许多图像测量（例如来自基于网络的照片共享站点或长期视频记录）的大型问题实例，此类算法的计算成本可能很高。在本文中，我们证明 _∞ 三角剖分承认核心集近似方案，该方案寻找输入数据的小型代表性子集（称为核心集）。核心集具有特殊属性，即核心集上 _∞ 解的误差在距全局最小值的已知范围内。我们为核心集算法建立了必要的数学基础，具体来说，通过制定算法的停止标准并证明生成的核心集给出了所需的近似精度。对于大规模三角测量问题，我们的方法提供了理论上合理的近似解。迭代直至收敛，我们的核心集算法也保证达到真正的最优值。在实际数据集上，我们表明我们的技术实际上可以比当前方法更快地实现全局最小化。