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 Gr\"obner 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 Gr\"obner 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.

中文翻译：

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通过隐藏变量计算稳定的基于结果的最小求解器

许多计算机视觉应用需要对相机几何结构进行稳健有效的估计。稳健估计通常基于从最少数量的输入数据测量解决相机几何问题，即在 RANSAC 风格的框架中解决最少的问题。最小的问题通常会导致复杂的多项式方程组。解决此类系统的现有最先进方法要么基于近年来已被广泛研究和优化的 Gr\"obner 基和动作矩阵方法，要么最近提出的基于稀疏结果计算的方法在本文中，我们研究了一种有趣的基于稀疏结果的替代方法，通过隐藏一个变量来求解多项式方程的稀疏系统。这种方法会导致比动作矩阵和基于额外变量稀疏结果的方法更大的特征值问题；然而，它不需要计算可能在数值上不稳定的大矩阵的逆或消去。所提出的方法包括对标准稀疏结果算法的多项改进，正如我们在几个有趣的计算机视觉问题上所展示的那样，这显着提高了基于隐藏变量结果的求解器的效率和稳定性。我们表明，对于所研究的问题，我们的基于稀疏结果的方法导致比最先进的基于 Gr\"obner 基的求解器以及现有的基于稀疏结果的求解器更稳定的求解器，尤其是在接近临界配置。