A Framework for Reducing the Complexity of Geometric Vision Problems and its Application to Two-View Triangulation with Approximation Bounds

2025-03-11

AI Summary

This paper addresses the high computational complexity of two-view triangulation in multiview geometry. We propose a reweighted projection error optimization framework that theoretically derives an optimal weighting strategy, reducing the conventional sextic polynomial minimization to a quadratic form amenable to closed-form analytical solution. To our knowledge, this is the first application of reweighted least squares to triangulation, achieving a favorable trade-off between efficiency and accuracy: geometric consistency is preserved while computation speed is significantly improved. We derive a theoretical upper bound on the approximation error, guaranteeing solution reliability. Extensive experiments on real-world datasets demonstrate that the proposed method achieves accuracy comparable to globally optimal solutions, with rigorously provable error control.

Technology Category

• Computer Vision: Learning & Optimization for CV
• Machine Learning: Multi-instance/Multi-view Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

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.

Problem

• Reduces computational complexity in geometric vision problems.
• Simplifies two-view triangulation via cost function reweighting.
• Provides closed-form solutions with strong geometric accuracy.

Innovation

• Reduces complexity via cost function reweighting
• Simplifies triangulation to degree two polynomial
• Provides closed-form solution with geometric accuracy