基于张量的同步与块三焦点张量的低秩性
Tensor-Based Synchronization and the Low-Rankness of the Block Trifocal Tensor
AI总结：
该研究建立块三焦点张量的显式Tucker分解，证明其低多线性秩特性可用于无噪声下的相机位姿恢复，提出基于高阶奇异值分解的同步算法，在真实数据集上显著提升位置估计精度，为同步问题提供高阶交互的新思路。
摘要：
三焦点张量构成的块张量为场景的三视图几何提供了关键的几何信息。其背后的同步问题旨在从块三焦点张量中恢复相机位姿（在全局变换下的位置与朝向）。我们建立了该张量的显式Tucker分解，发现在合适的缩放条件下，其多线性秩低至(6,4,4)，且与相机数量无关。我们证明了在无噪声情况下，该秩约束为相机恢复提供了充足的信息。受此约束启发，我们提出了一种基于块三焦点张量高阶奇异值分解的同步算法。在真实数据集上与当前最优的全局同步方法进行的实验对比表明，该算法有望显著提升位置估计精度。总体而言，这项工作表明，除了常用的基于成对关系的方法外，还可以利用同步问题中的高阶交互来提升性能。
英文摘要：
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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