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.

We present a method for solving two minimal problems for relative camera pose estimation from three views, which are based on three view correspondences of (i) three points and one line and the novel case of (ii) three points and two lines through two of the points. These problems are too difficult to be efficiently solved by the state of the art Gröbner basis methods. Our method is based on a new efficient homotopy continuation (HC) solver framework MINUS, which dramatically speeds up previous HC solving by specializing hc methods to generic cases of our problems. We characterize their number of solutions and show with simulated experiments that our solvers are numerically robust and stable under image noise, a key contribution given the borderline intractable degree of nonlinearity of trinocular constraints. We show in real experiments that (i) sift feature location and orientation provide good enough point-and-line correspondences for three-view reconstruction and (ii) that we can solve difficult cases with too few or too noisy tentative matches, where the state of the art structure from motion initialization fails.

This paper presents the Brown Pedestrian Odometry Dataset (BPOD) for benchmarking visual odometry algorithms on data from head-mounted sensors. This dataset was captured with stereo and RGB streams from RealSense cameras with rolling and global shutters in 12 diverse indoor and outdoor locations on Brown University's campus. Its associated ground-truth trajectories were generatedfrom third-person videos that documented the recorded pedestrians' positions relative to stick-on markers placed along their paths. We evaluate the performance of canonical approaches representative of direct, feature-based, and learning-based visual odometry methods on BPOD. Our finding is that current methods which are successful on other benchmarks fail on BPOD. The failure modes correspond in part to rapid pedestrian rotation, erratic body movements, etc. We hope this dataset will play a significant role in the identification of these failure modes and in the design, development, and evaluation of pedestrian odometry algorithms.

Systems of polynomial equations arise frequently in computer vision, especially in multiview geometry problems. Traditional methods for solving these systems typically aim to eliminate variables to reach a univariate polynomial, e.g., a tenth-order polynomial for 5-point pose estimation, using clever manipulations, or more generally using Grobner basis, resultants, and elimination templates, leading to successful algorithms for multiview geometry and other problems. However, these methods do not work when the problem is complex and when they do, they face efficiency and stability issues. Homotopy Continuation (HC) can solve more complex problems without the stability issues, and with guarantees of a global solution, but they are known to be slow. In this paper we show that HC can be parallelized on a GPU, showing significant speedups up to 56 times on polynomial benchmarks. We also show that GPU-HC can be generically applied to a range of computer vision problems, including 4-view triangulation and trifocal pose estimation with unknown focal length, which cannot be solved with elimination template but they can be efficiently solved with HC. GPU-HC opens the door to easy formulation and solution of a range of computer vision problems.

Relative pose estimation using the 5-point or 7-point Random Sample Consensus (RANSAC) algorithms can fail even when no outliers are present and there are enough inliers to support a hypothesis. These cases arise due to numerical instability of the 5- and 7-point minimal problems. This paper characterizes these instabilities, both in terms of minimal world scene configurations that lead to infinite condition number in epipolar estimation, and also in terms of the related minimal image feature pair correspondence configurations. The instability is studied in the context of a novel framework for analyzing the conditioning of minimal problems in multiview geometry, based on Riemannian manifolds. Experiments with synthetic and real-world data reveal that RANSAC does not only serve to filter out outliers, but RANSAC also selects for well-conditioned image data, sufficiently separated from the ill-posed locus that our theory predicts. These findings suggest that, in future work, one could try to accelerate and increase the success of RANSAC by testing only well-conditioned image data.

A key bottleneck in the use of Multiview Stereo (MVS) to produce high quality reconstructions is the gaps arising from textureless, shaded areas and lack of fine-scale detail. Shape-from-Shading (SfS) has been used in conjunction with MVS to obtain fine-scale detail and veridical reconstruction in the gap areas. The similarity metric that gauges candidate correspondences is critical to this process, typically a combination of photometric consistency and brightness gradient constancy. Two observations motivate this paper. First, brightness gradient constancy can be erroneous due to foreshortening. Second, the standard ZSSD/NCC patchwise photometric consistency measures when applied to shaded areas is, to a first-order approximation, a calculation of brightness gradient differences, which can be subject to foreshortening. The paper proposes a novel trinocular differential photometric consistency that constrains the brightness gradients in three views so that the image gradient in one view is completely determined by the image gradients at corresponding points in the the other two views. The theoretical developments here advocate the integration of this new measure, whose viability in practice has been demonstrated in a set of illustrative numerical experiments.

We present a method for solving two minimal problems for relative camera pose estimation from three views, which are based on three view correspondences of (i) three points and one line and (ii) three points and two lines through two of the points. These problems are too difficult to be efficiently solved by the state of the art Grobner basis methods. Our method is based on a new efficient homotopy continuation (HC) solver, which dramatically speeds up previous HC solving by specializing HC methods to generic cases of our problems. We show in simulated experiments that our solvers are numerically robust and stable under image noise. We show in real experiment that (i) SIFT features provide good enough point-and-line correspondences for three-view reconstruction and (ii) that we can solve difficult cases with too few or too noisy tentative matches where the state of the art structure from motion initialization fails.