Algebraic Varieties in Multiple View Geometry
Algebraic Varieties in Multiple View Geometry
复制标题
多视图几何中的代数簇
DOI：
10.1007/3-540-61123-1_180
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
Kalle Åström
中科院分区：
文献类型：
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作者：
A. Heyden;Kalle Åström
在本文中，我们将探讨不同的代数簇和理想，可以从多视图几何与未校准的相机。自然描述符Vn是在n个不同投影下的图像。另一个描述符是变量Vb，它是由视图对之间的所有双线性形式生成的，并且由所有双线性形式消失的所有点组成。另一个描述符，多样性，Vt，是由视图的三元组之间的所有三线性形式生成的多样性。当n =3时，V是一个可约簇，其中一个分量对应于V，另一个分量对应于三焦平面。在理想的理论术语中，这被称为初等分解。进一步证明了当n =3时，V是由三个双线性和一个三线性生成的，当n ≥4时，V是由（2n）个双线性生成的。这表明四个图像是代数设置中的一般情况，因为Vt可以仅由双线性生成。
In this paper we will investigate the different algebraic varieties and ideals that can be generated from multiple view geometry with uncalibrated cameras. The natural descriptor,Vn, is the image ofinunderndifferent projections. However, we will show thatVnis not a variety.Another descriptor, the varietyVb, is generated by all bilinear forms between pairs of views and consists of all points inwhere all bilinear forms vanish. Yet another descriptor, the variety,Vt, is the variety generated by all trilinear forms between triplets of views. We will show that whenn=3,Vtis a reducible variety with one component corresponding toVband another corresponding to the trifocal plane. In ideal theoretic terms this is called a primary decomposition. This settles the discussion on the connection between the bilinearities and the trilinearities.Furthermore, we will show that whenn=3,Vtis generated by the three bilinearities and one trilinearity and whenn≥4,Vtis generated by the (2n) bilinearities. This shows that four images is the generic case in the algebraic setting, becauseVtcan be generated by just bilinearities.