Systems of polynomial equations arise frequently in computer vision, especially in multiview geometry problems. Traditional methods for solving these systems typically aim to eliminate variables to reach a univariate polynomial, e.g., a tenth-order polynomial for 5-point pose estimation, using clever manipulations, or more generally using Grobner basis, resultants, and elimination templates, leading to successful algorithms for multiview geometry and other problems. However, these methods do not work when the problem is complex and when they do, they face efficiency and stability issues. Homotopy Continuation (HC) can solve more complex problems without the stability issues, and with guarantees of a global solution, but they are known to be slow. In this paper we show that HC can be parallelized on a GPU, showing significant speedups up to 26 times on polynomial benchmarks. We also show that GPU-HC can be generically applied to a range of computer vision problems, including 4-view triangulation and trifocal pose estimation with unknown focal length, which cannot be solved with elimination template but they can be efficiently solved with HC. GPU-HC opens the door to easy formulation and solution of a range of computer vision problems.
翻译：多式方程式系统在计算机视野中经常出现,特别是在多视几何问题中。解决这些系统的传统方法通常旨在消除各种变量,以达到一个单一的单象形多元度,例如,对五点的十阶多元度构成估计,使用巧妙的操纵,或更一般地使用格罗布纳基数、结果和清除模板,导致多视几何和其他问题的成功算法。然而,当问题复杂时,这些方法无法奏效,当它们面临效率和稳定性问题时,它们面临问题。智障Csustory(HC)可以解决更复杂的问题,而没有稳定性问题,并且有全球解决办法的保证,但众所周知,这些变量是缓慢的。在本文中,我们表明,在GPU上可以同时使用十级的十级多级组合,显示在多视几何和其他问题上可高达26次的大幅超速。我们还表明,GPU-HC可以被通用地应用于一系列计算机视觉问题,包括四视三角和三角相向不明的波形估测问题,这些问题无法通过消除模板加以解决加以解决,但可以有效地与HC的门形成一个容易的解决方案。GPUPI-HC-HC-HG-HG-HL-G-G-G-G-G-G-G-G-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-S-L-T-T-T-S-S-S-S-L-S-S-S-S-S-S-S-S-S-S-S-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-
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