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

arXiv: https://arxiv.org/abs/2310.02719 · HTML: https://mlsanity.com/paper/2310.02719/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 65 references that are in the map.
- First posted on arXiv 4 Oct 2023.

## Builds on (most cited first, 11 of 11)

- [MAGSAC++, a fast, reliable and accurate robust estimator](https://mlsanity.com/paper/1912.05909/) (2019)
- [A Survey of Structure from Motion](https://mlsanity.com/paper/1701.08493/) (2017)
- [Neural-Guided RANSAC: Learning Where to Sample Model Hypotheses](https://mlsanity.com/paper/1905.04132/) (2019)
- [Algebraic Characterization of Essential Matrices and Their Averaging in Multiview Settings](https://mlsanity.com/paper/1904.02663/) (2019)
- [PL1P -- Point-line Minimal Problems under Partial Visibility in Three Views](https://mlsanity.com/paper/2003.05015/) (2020)
- [On the Instability of Relative Pose Estimation and RANSAC's Role](https://mlsanity.com/paper/2112.14651/) (2021)
- [Snapshot of Algebraic Vision](https://mlsanity.com/paper/2210.11443/) (2022)
- [GPU-Based Homotopy Continuation for Minimal Problems in Computer Vision](https://mlsanity.com/paper/2112.03444/) (2021)
- [Machine learning the real discriminant locus](https://mlsanity.com/paper/2006.14078/) (2020)
- [Critical configurations for two projective views, a new approach](https://mlsanity.com/paper/2112.05074/) (2021)
- [The Chow Form of the Essential Variety in Computer Vision](https://mlsanity.com/paper/1604.04372/) (2016)

## Built on by (most cited first, 1 of 1)

- [Higher-Order Group Synchronization](https://mlsanity.com/paper/2505.21932/) (2025)

## Abstract (arXiv)

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.