A Framework for Reducing the Complexity of Geometric Vision Problems and its Application to Two-View Triangulation with Approximation Bounds
This work addresses a bottleneck in Structure-from-Motion pipelines for computer vision researchers, offering a more efficient method for two-view triangulation, though it is incremental as it builds on existing optimization techniques.
The paper tackles the computational complexity of geometric vision problems by introducing a framework that reduces the degree of polynomials in triangulation from six to two through cost function reweighting, achieving a closed-form solution with strong geometric accuracy and theoretical error bounds.
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.