通过隐藏变量计算稳定的基于结果的极小解算器
Computing stable resultant-based minimal solvers by hiding a variable
摘要：
许多计算机视觉应用需要鲁棒且高效的相机几何形状估计。鲁棒估计通常基于从最小数量的输入数据测量中解决摄像机几何问题，即，解决最小的问题，在一个RANSAC风格的框架。最小问题往往导致复杂的多项式方程组。现有的最先进的方法来解决这样的系统是基于格罗布纳基和行动矩阵的方法，这已被广泛研究和优化，在最近几年或最近提出的方法的基础上稀疏的结果计算使用一个额外的变量。在本文中，我们研究了一个有趣的替代稀疏结果为基础的方法来解决稀疏多项式方程组通过隐藏一个变量。这种方法比基于作用矩阵和额外变量稀疏结果的方法导致更大的特征值问题;然而，它不需要计算可能在数值上不稳定的大型矩阵的逆或消除。所提出的方法包括几个改进的标准稀疏结果算法，显着提高了效率和稳定性的隐变量结果为基础的求解器，我们证明了几个有趣的计算机视觉问题。我们表明，对于所研究的问题，我们的稀疏结果为基础的方法导致更稳定的求解器比国家的最先进的格罗布纳基为基础的求解器，以及现有的稀疏结果为基础的求解器，特别是在接近临界配置。我们的新方法可以完全自动化，并纳入现有的工具，自动生成高效的最小求解器。
英文摘要：
Many computer vision applications require robust and efficient estimation of camera geometry. The robust estimation is usually based on solving camera geometry problems from a minimal number of input data measurements, i.e., solving minimal problems, in a RANSAC-style framework. Minimal problems often result in complex systems of polynomial equations. The existing state-of-the-art methods for solving such systems are either based on Gröbner bases and the action matrix method, which have been extensively studied and optimized in the recent years or recently proposed approach based on a sparse resultant computation using an extra variable. In this paper, we study an interesting alternative sparse resultant-based method for solving sparse systems of polynomial equations by hiding one variable. This approach results in a larger eigenvalue problem than the action matrix and extra variable sparse resultant-based methods; however, it does not need to compute an inverse or elimination of large matrices that may be numerically unstable. The proposed approach includes several improvements to the standard sparse resultant algorithms, which significantly improves the efficiency and stability of the hidden variable resultant-based solvers as we demonstrate on several interesting computer vision problems. We show that for the studied problems, our sparse resultant based approach leads to more stable solvers than the state-of-the-art Gröbner bases-based solvers as well as existing sparse resultant-based solvers, especially in close to critical configurations. Our new method can be fully automated and incorporated into existing tools for the automatic generation of efficient minimal solvers.
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