🤖 AI Summary
This work addresses the failure of 5-point and 7-point RANSAC-based relative pose estimation in multiview geometry—even under outlier-free conditions and sufficient data—identifying inherent numerical instability in minimal problems as the root cause. To tackle this, we propose a unified conditioning analysis framework that integrates computational algebra (Gröbner bases, elimination theory) with Riemannian geometry (condition number definition on manifolds), enabling the first computable modeling and scene-level predictability of condition numbers for the 5- and 7-point problems. Theoretically, we reveal a previously unrecognized mechanism: RANSAC implicitly favors well-conditioned image configurations due to its sampling and scoring behavior. Extensive experiments confirm that this implicit bias is essential for RANSAC’s success in outlier-free settings and demonstrate substantial improvements in pose estimation robustness and reliability across diverse geometric scenarios.
📝 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.
