Hongyi Fan, Joe Kileel, and Benjamin Kimia, “Condition numbers in multiview geometry, instability in relative pose estimation, and RANSAC.” arXiv, October 2023 [Online]. Available at: http://arxiv.org/abs/2310.02719. [Accessed: October 27, 2023]
@misc{fan_condition_2023,
title = {Condition numbers in multiview geometry, instability in relative pose estimation, and {RANSAC}},
url = {http://arxiv.org/abs/2310.02719},
arxivdoi = {10.48550/arXiv.2310.02719},
urldate = {2023-10-27},
publisher = {arXiv},
author = {Fan, Hongyi and Kileel, Joe and Kimia, Benjamin},
month = oct,
year = {2023},
note = {arXiv:2310.02719 [cs, math]},
keywords = {Computer Science - Computer Vision and Pattern Recognition, Mathematics - Numerical Analysis},
freepdf = {https://arxiv.org/pdf/2310.02719.pdf},
tldr = {Argues that the 5-point and 7-point (compute essential, and fundamental matrix, respectively from image correspondences) algorithms may be numerically unstable even in cases with no outliers. Then RANSAC not only filters outliers, but also tends towards selecting data points such that condition numbers are well-behaved.}
}
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.
tl;dr: Argues that the 5-point and 7-point (compute essential, and fundamental matrix, respectively from image correspondences) algorithms may be numerically unstable even in cases with no outliers. Then RANSAC not only filters outliers, but also tends towards selecting data points such that condition numbers are well-behaved.
Condition numbers in multiview geometry, instability in relative pose estimation, and RANSAC.
arXiv: http://doi.org/10.48550/arXiv.2310.02719
h/t Amy Tabbpdf: https://arxiv.org/pdf/2310.02719.pdf