用张量分解实现四焦点张量同步，突破传统相机重建瓶颈
QuadSync: Quadrifocal Tensor Synchronization via Tucker Decomposition
- 构建块四焦点张量并利用Tucker分解提取相机矩阵
- 首次实现四焦点张量同步，多视角重建精度显著提升
- 适合三维重建与计算机视觉研究者，尤其关注高阶几何建模
在运动恢复结构中，四焦点张量比成对的本质矩阵蕴含更多信息，但长期被认为不实用且仅具理论价值。本文提出新框架，从对应四焦点张量集合中恢复 $n$ 个相机。构造块四焦点张量，证明其可进行Tucker分解，因子矩阵为堆叠的相机矩阵，且多线性秩恒为 (4, 4, 4, 4)，与 $n$ 无关。开发首个基于Tucker分解、交替方向乘子法和迭代加权最小二乘的四焦点张量同步算法。进一步揭示块四焦点、三焦点和二焦点张量间的关联，提出联合同步三类张量的算法。数值实验在现代数据集上验证方法有效性，表明高阶信息在同步中的潜力与重要性。
原文摘要 · Abstract (English)
In structure from motion, quadrifocal tensors capture more information than their pairwise counterparts (essential matrices), yet they have often been thought of as impractical and only of theoretical interest. In this work, we challenge such beliefs by providing a new framework to recover $n$ cameras from the corresponding collection of quadrifocal tensors. We form the block quadrifocal tensor and show that it admits a Tucker decomposition whose factor matrices are the stacked camera matrices, and which thus has a multilinear rank of (4,~4,~4,~4) independent of $n$. We develop the first synchronization algorithm for quadrifocal tensors, using Tucker decomposition, alternating direction method of multipliers, and iteratively reweighted least squares. We further establish relationships between the block quadrifocal, trifocal, and bifocal tensors, and introduce an algorithm that jointly synchronizes these three entities. Numerical experiments demonstrate the effectiveness of our methods on modern datasets, indicating the potential and importance of using higher-order information in synchronization.
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