揭示三组相机矩阵的几何兼容性，提出更完整的代数约束。
Multiprojective Geometry of Compatible Triples of Fundamental and Essential Matrices
- 通过计算多重度与多重齐次理想，刻画兼容基础矩阵三元组的代数结构。
- 发现一组简单的四次约束，能完整描述兼容基础矩阵三元组的代数关系。
- 结果对本质矩阵也适用，适合研究几何视觉中矩阵兼容性的研究人员。
我们通过计算多重度和多重齐次消失理想，刻画了兼容基础矩阵三元组的代数簇。这回答了Bråtelund和Rydell最近提出的问题中的首个有趣情形。我们的结果改进了现有几何计算机视觉文献中已知的代数约束集——这些旧约束均不完整（未能生成消失理想），且有时对矩阵三元组的缩放施加了限制性假设。我们的讨论还扩展至更一般的兼容性簇，其多重齐次消失理想尚不明确。一个关键新发现是：存在一组简单的四次约束，其在兼容基础矩阵三元组上恒为零。这些四次约束在本质矩阵场景中同样重要：结合已有约束，可局部刻画出兼容本质矩阵三元组的代数簇。
原文摘要 · Abstract (English)
We characterize the variety of compatible fundamental matrix triples by computing its multidegree and multihomogeneous vanishing ideal. This answers the first interesting case of a question recently posed by Bråtelund and Rydell. Our result improves upon previously discovered sets of algebraic constraints in the geometric computer vision literature, which are all incomplete (as they do \emph{not} generate the vanishing ideal) and sometimes make restrictive assumptions about how a matrix triple should be scaled. Our discussion touches more broadly on generalized compatibility varieties, whose multihomogeneous vanishing ideals are much less well understood. One of our key new discoveries is a simple set of quartic constraints vanishing on compatible fundamental matrix triples. These quartics are also significant in the setting of essential matrices: together with some previously known constraints, we show that they locally cut out the variety of compatible essential matrix triples.
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