We consider Galois/monodromy groups arising in computer vision applications, with a view towards building more efficient polynomial solvers. The Galois/monodromy group allows us to decide when a given problem decomposes into algebraic subproblems, and whether or not it has any symmetries. Tools from numerical algebraic geometry and computational group theory allow us to apply this framework to classical and novel reconstruction problems. We consider three classical cases--3-point absolute pose, 5-point relative pose, and 4-point homography estimation for calibrated cameras--where the decomposition and symmetries may be naturally understood in terms of the Galois/monodromy group. We then show how our framework can be applied to novel problems from absolute and relative pose estimation. For instance, we discover new symmetries for absolute pose problems involving mixtures of point and line features. We also describe a problem of estimating a pair of calibrated homographies between three images. For this problem of degree 64, we can reduce the degree to 16; the latter better reflecting the intrinsic difficulty of algebraically solving the problem. As a byproduct, we obtain new constraints on compatible homographies, which may be of independent interest.
翻译:我们考虑了计算机视觉应用中产生的伽洛瓦/摩诺德罗米群体,目的是建立更有效率的多元溶液。伽洛瓦/莫诺德罗米群体允许我们决定一个特定问题何时分解成代数子问题,以及它是否具有任何对称性。数字代数几何和计算组理论的工具允许我们将这一框架应用于传统和新颖的重建问题。我们考虑三个典型案例 -- -- 3点绝对面、5点相对面和4点对校准相机的同系估计 -- -- 可以自然地从伽洛瓦/摩多罗米群体的角度理解分解和对称问题。然后我们展示我们的框架如何从绝对和相对面的估测中应用到新问题。举例来说,我们发现了绝对构成涉及点和线性混合物问题的新对称。我们还描述了在三种图像之间估计一对校准同系的问题。关于这个程度64的问题,我们可以将分解度降低到16度;后一种框架可以更好地从绝对和相对的估量的角度反映我们所要解决的内含利息问题的内在困难。