Many computer vision applications require robust and efficient estimation of camera geometry from a minimal number of input data measurements, i.e., solving minimal problems in a RANSAC framework. Minimal problems are usually formulated as complex systems of sparse polynomials. The systems usually are overdetermined and consist of polynomials with algebraically constrained coefficients. Most state-of-the-art efficient polynomial solvers are based on the action matrix method that has been automated and highly optimized in recent years. On the other hand, the alternative theory of sparse resultants and Newton polytopes has been less successful for generating efficient solvers, primarily because the polytopes do not respect the constraints on the coefficients. Therefore, in this paper, we propose a simple iterative scheme to test various subsets of the Newton polytopes and search for the most efficient solver. Moreover, we propose to use an extra polynomial with a special form to further improve the solver efficiency via a Schur complement computation. We show that for some camera geometry problems our extra polynomial-based method leads to smaller and more stable solvers than the state-of-the-art Grobner basis-based solvers. The proposed method can be fully automated and incorporated into existing tools for automatic generation of efficient polynomial solvers. It provides a competitive alternative to popular Grobner basis-based methods for minimal problems in computer vision. We also study the conditions under which the minimal solvers generated by the state-of-the-art action matrix-based methods and the proposed extra polynomial resultant-based method, are equivalent. Specifically we consider a step-by-step comparison between the approaches based on the action matrix and the sparse resultant, followed by a set of substitutions, which would lead to equivalent minimal solvers.

翻译：许多计算机视觉应用需要从最少输入数据测量中稳健高效地估计相机几何，即在RANSAC框架下求解最小问题。最小问题通常被表述为稀疏多项式构成的复杂系统，这类系统通常超定且包含系数受代数约束的多项式。现有最先进的求解器大多基于近年来已实现自动化与高度优化的作用矩阵方法。另一方面，稀疏结式与牛顿多面体的替代理论在生成高效求解器方面成效有限，主要原因是多面体无法顾及系数约束。因此，本文提出一种简单的迭代方案来测试牛顿多面体的不同子集，并搜索最高效的求解器。此外，我们提出利用具有特殊形式的额外多项式，通过舒尔补计算进一步提升求解器效率。对于某些相机几何问题，我们的额外多项式方法比基于最先进的Grobner基方法生成的求解器更小且更稳定。所提方法可完全自动化，并集成到现有自动生成高效多项式求解器的工具中，为计算机视觉中的最小问题提供了一个有竞争力的替代方案，以替代流行的基于Grobner基的方法。我们还研究了最先进的基于作用矩阵的方法与所提出的额外多项式结式方法生成的求解器等价的条件。具体而言，我们对基于作用矩阵与稀疏结式的方法进行逐步对比，并通过一系列替换推导出等价的求解器。