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

TL;DR

This work addresses the computational burden of geometric vision problems, focusing on triangulation, by introducing a framework that reduces problem complexity through targeted reweighting of the objective and a coordinate diagonalization of the constraints. Applied to two-view triangulation, the method lowers the ED-degree from $6$ to $2$ and yields a closed-form solution, with theoretically derived optimal weights and provable approximation bounds. The approach is validated on real data and shown to be competitive with, and sometimes superior to, established baselines, especially when cameras have parallel optical axes. The framework is general and promises extensions to higher-view problems, potentially simplifying otherwise intractable algebraic optimization tasks in multiview geometry.

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.

Figures (5)

• Figure 1: The optimal triangulation problem can have up to three local minima and requires finding the roots of a degree-6 univariate polynomial, following Hartley-Sturm hartley1997triangulation. We propose a weighting of the cost function that lowers the degree to 2, yielding a unique minimum. The plot shows a specific example where Hartley-Sturm's method has to choose between two local minima. Here, we plot the lowest cost value per epipolar plane, parameterised by an angle from the principal axis of one camera (see hartley1997triangulation for details). The costs are rescaled to have the same minimum and maximum value for ease of viewing.
• Figure 2: 2D errors over correspondences from randomly sampled image pairs from the Pantheon dataset, when solving the triangulation problem using different methods. The methods perform very similarly, but our weighted method is slightly worse than the others. Left: Distance to ground truth projections; Right: Distance to measured 2D points.
• Figure 3: The eigenvalue ratios in our randomly sampled set of 5000 image pairs from the Pantheon dataset.
• Figure 4: Evaluation of the error bounds \ref{['eq: upper bound']} and \ref{['eq: bounds']} under different amounts of relative rotation between cameras. For cameras with close to the same viewing direction, the bounds are sharp as predicted by the theory. The markers are scaled (logarithmically) by the eigenvalue ratio \ref{['eq:eigRatio']}. We subsample 100 random correspondences to plot, for ease of viewing.
• Figure 5: Evaluation of the error bounds \ref{['eq: upper bound']} and \ref{['eq: bounds']} under a large amount of noise on the 2D correspondences. The markers are scaled (logarithmically) by the eigenvalue ratio \ref{['eq:eigRatio']} as in Figure \ref{['fig:bounds']}. We subsample 100 random correspondences to plot, for ease of viewing. At this point Sampson fails to approximate the optimal errors well, but all of the correspondences would anyway be classified as outliers, so practically this failure is not relevant.

Theorems & Definitions (16)

• Example 2.1: Triangulation
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Remark 3.3
• Theorem 4.1
• proof
• Lemma 4.2
• proof