通过重加权降低三维重建计算复杂度，实现高精度闭式解
A Framework for Reducing the Complexity of Geometric Vision Problems and its Application to Two-View Triangulation with Approximation Bounds
- 用重加权重构代价函数，将六次多项式降为二次
- 闭式解误差小于标准方法，理论误差界可证明
- 适合需要实时三维重建的视觉系统开发者
本文提出一种通过针对性重加权代价函数来降低几何视觉问题计算复杂度的新框架。三角化——从多视图中噪声2D投影估计3D点——是多视图几何与运动恢复结构（SfM）中的基础问题。我们将该框架应用于双视图情形，证明通过代价函数重加权，原本需求解六次单变量多项式的最优三角化，可简化为二次多项式，从而获得闭式解，同时保持强几何精度。我们推导出最优权重策略，建立近似误差的理论边界，并在真实数据上验证了该方法优于标准方法的有效性。尽管聚焦于双视图三角化，该框架可推广至其他几何视觉问题。
原文摘要 · Abstract (English)
In this paper, we present a new framework for reducing the computational complexity of geometric vision problems through targeted reweighting of the cost functions used to minimize reprojection errors. Triangulation - the task of estimating a 3D point from noisy 2D projections across multiple images - is a fundamental problem in multiview geometry and Structure-from-Motion (SfM) pipelines. We apply our framework to the two-view case and demonstrate that optimal triangulation, which requires solving a univariate polynomial of degree six, can be simplified through cost function reweighting reducing the polynomial degree to two. This reweighting yields a closed-form solution while preserving strong geometric accuracy. We derive optimal weighting strategies, establish theoretical bounds on the approximation error, and provide experimental results on real data demonstrating the effectiveness of the proposed approach compared to standard methods. Although this work focuses on two-view triangulation, the framework generalizes to other geometric vision problems.
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