Variety
Variety
复制标题
种类
DOI：
--
复制
发表时间：
--
期刊：
影响因子：
--
通讯作者：
F. Nightingale
中科院分区：
文献类型：
--
作者：
F. Nightingale
本文探讨多视图几何中的以下问题。考虑一个相机由从一个点（焦点）投影到一个平面组成。如果我们有三个固定的相机，我们可以构建如下应用：给定相机1和2的投影平面中的两条直线$l_1$、$l_2$，每一条直线通过将其与自身焦点相连可确定一个平面；将这个平面与相机3的投影平面相交，就会得到另一条直线$l_3$。那么这个应用就是从$\mathbb{P}^2\times\mathbb{P}^2$到$\mathbb{P}^2$的$(l_1, l_2)\mapsto l_3$。这等价于一个张量，称为三焦张量。现在，通过一种不同的代数几何构造，可以定义三焦簇。然后，本文给出了这个簇的理想的极小生成元。主要结果是该理想由大约2100个3到6次的多项式极小生成。从某种意义上说，这改进了2010年的一个先前结果，该结果提供了在集合论意义下切出该簇的方程，但没有生成理想。
This article deals with the following problem in multiview geometry. Consider that a camera consists on projecting from a point (focus) to a plane. If we have three fixed cameras we can construct the following application: given two lines l1, l2 in the projecting planes of cameras 1 and 2, each one defines a plane upon linking it to its focus; intersect this with the projecting plane of camera 3, and you obtain another line l3. The application is then (l1, l2) 7→ l3, from P2 × P2 to P2. This is equivalent to a tensor, called a trifocal tensor. Now, via a different, algebro-geometric construction, one can define the trifocal variety. Then, this article provides minimal generators of the ideal of this variety. The main result is that the ideal is minimally generated by about 2100 polynomials of degrees 3 to 6. This improves, in a sense, a previous result of 2010 providing equations that cut out the variety set-theoretically, but that do not generate the ideal.