证明了多视角理想的一组多项式构成通用格罗布纳基。
Universal Gröbner Bases of (Universal) Multiview Ideals
- 用黄和拉森的判据，构造出通用格罗布纳基。
- 对无限多个理想适用，通过对称约化与归纳法实现。
- 揭示了多视角理想依赖的组合结构——拟阵。
多视角理想源于针孔相机成像几何，而通用多视角理想是未知相机情形的对应物。我们利用黄和拉森提出的判据，证明了一组自然的多项式构成了这两类理想的标准格罗布纳基，并给出了该判据在本设定下的证明。通过对称约化与归纳法，该方法可推广至无限多个理想的情形。此外，我们还显式描述了该方法所依赖的拟阵结构，其背景为多视角理想。
原文摘要 · Abstract (English)
Multiview ideals arise from the geometry of image formation in pinhole cameras, and universal multiview ideals are their analogs for unknown cameras. We prove that a natural collection of polynomials form a universal Gröbner basis for both types of ideals using a criterion introduced by Huang and Larson, and include a proof of their criterion in our setting. Symmetry reduction and induction enable the method to be deployed on an infinite family of ideals. We also give an explicit description of the matroids on which the methodology depends, in the context of multiview ideals.
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