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 also to select for well-conditioned image data, as predicted by our theory.

翻译：本文利用计算代数与黎曼几何工具，建立了一个分析多视图几何中最小问题数值条件的一般框架。研究动机源于以下事实：基于标准五点或七点随机抽样一致性（RANSAC）算法的相对位姿估计，即使在没有外点且数据足以支撑假设的情况下仍可能失败。我们认为，这些情况源于五点和七点最小问题的固有非稳定性。我们应用该框架从两方面刻画这种不稳定性：一方面分析导致无限条件数的真实场景，另一方面直接描述病态图像数据。该方法可在求解最小问题前提供评估条件数的计算测试。最后，合成数据与真实数据实验表明，RANSAC不仅能够剔除外点，还能如理论预测般筛选出良态图像数据。