Revisiting Sampson Approximations for Geometric Estimation Problems

TL;DR

This work revisits the Sampson error as a fast linearized surrogate for the true geometric residual in polynomial constraints, deriving explicit bounds on when the approximation is tight under mild assumptions. It extends the framework to multiple constraints with covariance weighting and addresses general cases via Moore–Penrose inverses, including practical strategies for choosing a full-rank constraint subset when many constraints are present. The authors validate the theory through extensive experiments across two-view and three-view pose estimation, vanishing-point refinement, and 2D/3D reprojection with uncertainties, demonstrating that Sampson-based objectives yield accurate results with much lower computational cost than full nonlinear optimization. Overall, the paper provides both theoretical guarantees and empirical guidance for deploying Sampson approximations in robust geometric estimation pipelines.

Abstract

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.

Figures (6)

• Figure 1: The model $C(x,y)= x^2+2y^2-4 =0$ is the ellipse on the top left, and the data points are $\boldsymbol{z}_1$ and $\boldsymbol{z}_2$. On the top right, the gray surface is the graph $(x,y,C(x,y))$, the orange and purple curves are the level sets $C(x,y)=C(\boldsymbol{z}_1)$ and $C(x,y)=C(\boldsymbol{z}_2)$ respectively. In the bottom right the blue planes are tangent to the surface at $(\boldsymbol{z}_1,C(\boldsymbol{z}_1))$ and $(\boldsymbol{z}_2,C(\boldsymbol{z}_2))$ respectively. The orange and purple lines are the linearized constraints for $\boldsymbol{z}_1$ and $\boldsymbol{z}_2$ respectively. We keep this color convention for the linearized constraints on the bottom left, and represent their normals in red. The Sampson approximations for $\boldsymbol{z}_1$ and $\boldsymbol{z}_2$ are the red points $\boldsymbol{z}_1+\boldsymbol{\varepsilon}_1^S$ and $\boldsymbol{z}_2+\boldsymbol{\varepsilon}_2^S$, and the minimizers of the geometric error are depicted in green. Since $\boldsymbol{z}_1$ is in the gray region (obtained from Proposition \ref{['prop: eG bound']}), its Sampson approximation is better than the approximation for $\boldsymbol{z}_2$.
• Figure 2: Regions obtained for \ref{['ex:ellipse']}. On the left, the constraint $x_1^2+2x_2^2-4=0$ is depicted in blue, the purple region is obtained from \ref{['eq:eG_relaxed_bound']} and it is contained in the orange region coming from \ref{['eq:eG_bound']}. On the right, the level sets for the ratio $\|\boldsymbol{\varepsilon}^G\|/\|\boldsymbol{\varepsilon}^S\|$, and the constraint depicted in blue. Observe that, although the ratio changes, it is bounded by 2 for every point in the colored regions.
• Figure 3: Approximation gap for two-view relative pose. Comparison with optimal triangulation lindstrom2010triangulation. The unit is in pixels. Here $\mathcal{E}$ refers to either the Sampson or the symmetric epipolar error.
• Figure 4: Approximation error against $\rho|C(\boldsymbol{z})|/\|J\|^2$ for two-view relative pose. Figure shows a heatmap built from the inlier correspondences of $\sim$5k image pairs from the British Museum scene. Points that are either close to satisfying the epipolar constraint ($C(\boldsymbol{z})\approx 0$) or have low curvature ($\rho$) have smaller errors.
• Figure 5: Evaluation of the bounds for VP-line error. The lower bound $B_l$ and upper bound $B_u$ for the $\approx$350k VP-line pairs in the combined dataset. For illustration the pairs are sorted w.r.t. the tightness of the bound.

Theorems & Definitions (17)

• Remark 2.1
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Example 3.3
• Proposition 3.4
• proof
• Lemma A.1
• proof