Condition numbers in multiview geometry, instability in relative pose estimation, and RANSAC
This addresses instability issues in computer vision for researchers and practitioners, offering insights into when standard algorithms fail, but it is incremental as it builds on existing minimal problem analysis.
The paper tackles the problem of numerical instability in relative pose estimation using 5-point and 7-point RANSAC algorithms, even in outlier-free scenarios, by analyzing the condition numbers of minimal problems in multiview geometry. It introduces a framework to characterize instabilities and shows that RANSAC implicitly selects well-conditioned data, with experiments supporting this theory.
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.