Tensor-Based Synchronization and the Low-Rankness of the Block Trifocal Tensor

Daniel Miao, Gilad Lerman, Joe Kileel
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Abstract

The block tensor of trifocal tensors provides crucial geometric information on the three-view geometry of a scene. The underlying synchronization problem seeks to recover camera poses (locations and orientations up to a global transformation) from the block trifocal tensor. We establish an explicit Tucker factorization of this tensor, revealing a low multilinear rank of $(6,4,4)$ independent of the number of cameras under appropriate scaling conditions. We prove that this rank constraint provides sufficient information for camera recovery in the noiseless case. The constraint motivates a synchronization algorithm based on the higher-order singular value decomposition of the block trifocal tensor. Experimental comparisons with state-of-the-art global synchronization methods on real datasets demonstrate the potential of this algorithm for significantly improving location estimation accuracy. Overall this work suggests that higher-order interactions in synchronization problems can be exploited to improve performance, beyond the usual pairwise-based approaches.
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基于张量的同步和块状三焦张量的低空白度
三焦点张量的块张量提供了场景三视几何的关键几何信息。基本同步问题旨在从块三焦点张量中恢复摄像机姿态(全局变换前的位置和方向)。我们建立了该张量的显式塔克因子化,揭示了在适当的缩放条件下与摄像机数量无关的$(6,4,4)$低多线性秩。我们证明,在无噪声的情况下,这个秩约束为摄像机识别提供了足够的信息。该约束激发了一种基于块焦点张量的高阶奇异值分解的同步算法。在真实数据集上与最先进的全局同步方法进行的实验比较表明,这种算法具有显著提高位置估计精度的潜力。总之,这项工作表明,同步问题中的高阶交互作用可以被利用来提高性能,而不是通常的基于配对的方法。
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