Condition numbers in multiview geometry, instability in relative pose
  estimation, and RANSAC
Hongyi Fan Joe Kileel Benjamin Kimia
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.