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

Revisiting Sampson Approximations for Geometric Estimation Problems

Many problems in computer vision can be formulated as geometric estimation problems, i.e. given a collection of measurements (e.g. point correspondences) we wish to fit a model (e.g. an essential matrix) that agrees with our observations. This necessitates some measure of how much an observation ``agrees" with a given model. A natural choice is to consider the smallest perturbation that makes the observation exactly satisfy the constraints. However, for many problems, this metric is expensive or otherwise intractable to compute. The so-called Sampson error approximates this geometric error through a linearization scheme. For epipolar geometry, the Sampson error is a popular choice and in practice known to yield very tight approximations of the corresponding geometric residual (the reprojection error). In this paper we revisit the Sampson approximation and provide new theoretical insights as to why and when this approximation works, as well as provide explicit bounds on the tightness under some mild assumptions. Our theoretical results are validated in several experiments on real data and in the context of different geometric estimation tasks.