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Condition numbers in multiview geometry, instability in relative pose estimation, and RANSAC

Hongyi Fan, Joe Kileel, Benjamin Kimia

TL;DR

This work develops a general conditioning framework for minimal problems in multiview geometry, linking world-scene conditioning to the Jacobian of the image-data forward map and introducing discriminant-based tools to detect ill-posedness. It applies the framework to the $5$-point (essential matrix) and $7$-point (fundamental matrix) problems, deriving explicit condition-number formulas and geometric characterizations of ill-posed world scenes and image data via quadric surfaces and discriminants. The authors introduce X.5-point curves as practical proxies for conditioning in image space, with numerical and symbolic methods to compute them, and demonstrate that instability can affect RANSAC even in the absence of outliers. Experimental results on synthetic and real data show that poorly conditioned configurations correlate with large pose estimation errors and that RANSAC tends to favor well-conditioned image data, suggesting stability-aware improvements to robust estimation pipelines. Overall, the paper provides a rigorous, actionable view of numerical stability in two-view geometry and lays groundwork for stability-guided pose estimation strategies.

Abstract

In this paper, we introduce a general framework for analyzing the numerical conditioning of minimal problems in multiple view geometry, using tools from computational algebra and Riemannian geometry. Special motivation comes from the fact that relative pose estimation, based on standard 5-point or 7-point Random Sample Consensus (RANSAC) algorithms, can fail even when no outliers are present and there is enough data to support a hypothesis. We argue that these cases arise due to the intrinsic instability of the 5- and 7-point minimal problems. We apply our framework to characterize the instabilities, both in terms of the world scenes that lead to infinite condition number, and directly in terms of ill-conditioned image data. The approach produces computational tests for assessing the condition number before solving the minimal problem. Lastly, synthetic and real data experiments suggest that RANSAC serves not only to remove outliers, but in practice it also selects for well-conditioned image data, which is consistent with our theory.

Condition numbers in multiview geometry, instability in relative pose estimation, and RANSAC

TL;DR

This work develops a general conditioning framework for minimal problems in multiview geometry, linking world-scene conditioning to the Jacobian of the image-data forward map and introducing discriminant-based tools to detect ill-posedness. It applies the framework to the -point (essential matrix) and -point (fundamental matrix) problems, deriving explicit condition-number formulas and geometric characterizations of ill-posed world scenes and image data via quadric surfaces and discriminants. The authors introduce X.5-point curves as practical proxies for conditioning in image space, with numerical and symbolic methods to compute them, and demonstrate that instability can affect RANSAC even in the absence of outliers. Experimental results on synthetic and real data show that poorly conditioned configurations correlate with large pose estimation errors and that RANSAC tends to favor well-conditioned image data, suggesting stability-aware improvements to robust estimation pipelines. Overall, the paper provides a rigorous, actionable view of numerical stability in two-view geometry and lays groundwork for stability-guided pose estimation strategies.

Abstract

In this paper, we introduce a general framework for analyzing the numerical conditioning of minimal problems in multiple view geometry, using tools from computational algebra and Riemannian geometry. Special motivation comes from the fact that relative pose estimation, based on standard 5-point or 7-point Random Sample Consensus (RANSAC) algorithms, can fail even when no outliers are present and there is enough data to support a hypothesis. We argue that these cases arise due to the intrinsic instability of the 5- and 7-point minimal problems. We apply our framework to characterize the instabilities, both in terms of the world scenes that lead to infinite condition number, and directly in terms of ill-conditioned image data. The approach produces computational tests for assessing the condition number before solving the minimal problem. Lastly, synthetic and real data experiments suggest that RANSAC serves not only to remove outliers, but in practice it also selects for well-conditioned image data, which is consistent with our theory.
Paper Structure (44 sections, 18 theorems, 137 equations, 17 figures)

This paper contains 44 sections, 18 theorems, 137 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Minimal Problems for Relative Pose Estimation
  3. Essential matrices and the 5-point problem
  4. Fundamental matrices and the 7-point problem
  5. General Framework
  6. Spaces and maps
  7. Definition of condition number
  8. Definition of ill-posed locus
  9. Tool from computational algebra
  10. Main Theoretical Results
  11. Condition number formulas
  12. Ill-posed world scenes
  13. Ill-posed image data
  14. Relation Between Ill-Posed ness, Degeneracy and Criticality
  15. Computation of X.5-Point Curves
  16. ...and 29 more sections

Key Result

Proposition 3

The action of $G$ on the subset of $\mathcal{C}^{\times 2} \times (\mathbb{R}^3)^{\times 5}$ in eq:W-essential is smooth, free and proper. Therefore the quotient $\mathcal{W}$ is canonically a smooth manifold of dimension $20$.

Figures (17)

  • Figure 1: Typical relative pose estimation can fail catastrophically, even with a large number of correspondences (100 correspondences shown in the figure), all of which are inliers. (a) Ground-truth epipolar geometry. (b) Erroneous estimated epipolar geometry, from the 7-point algorithm and LO-RANSAC which also uses local refinement chum2003locally. The prime cause of such failure is numerical instability as shown in this paper.
  • Figure 2: (a) An ill-posed world scene in the calibrated case. Red and blue pyramids represent two cameras. Magenta points represent five world points. The green surface is a quadric surface satisfying the conditions of Theorem \ref{['thm:illposed-world']}. The view (b) shows an orange plane perpendicular to the baseline whose intersection with the quadric surface is a circle. (c) Another ill-posed world scene for the $5$-point problem, in the case where a plane perpendicular to the baseline intersect with the quadric surface in a line. View (d) shows the orange plane perpendicular to the baseline.
  • Figure 3: Example of an uncalibrated scene that is critical, in which all minimal subscenes are well-posed. Fix a ruled quadric surface (orange surface) as shown. Any $N$ points and $2$ camera centers (magenta points) drawn from the quadric will form a critical configuration by krames1941ermittlung. However generically, the baseline (magenta line) will not lie on the surface.
  • Figure 4: (a) To generate X.5-point curves on the second image, we can sweep the image columnwise and compute the intersection with vertical lines by solving \ref{['eq:P1P2-constraint']}, \ref{['eq:P3-constraint']}, \ref{['eq:Jac-drop']}. (b) To assess a candidate correspondence, we can scan just a neighborhood around the candidate point.
  • Figure 5: Ratio of unstable instances out of $3000$ random synthetic minimal problem instances at different noise levels $\sigma$ and error thresholds $\tau$ for: (a) fundamental matrices; and (b) essential matrices.
  • ...and 12 more figures

Theorems & Definitions (29)

  • Definition 1
  • Definition 2
  • Proposition 3
  • Lemma 4
  • Proposition 5
  • Lemma 6
  • Definition 7
  • Lemma 8
  • Definition 9
  • Definition 10
  • ...and 19 more