Relative pose estimation for RGB-D cameras is crucial in a number of applications. A typical approach relies on RANSAC to find a triplet pair of 3D point correspondences from which relative pose can be derived. A key aspect to this work ensures the geometric consistency of the triplet, i.e., pairwise distances between 3D points are preserved between the two views. Observe, however, that depth values are typically an order of magnitude less precise than feature locations, leading to large distance thresholds and admission of numerous false positives. This paper proposes that the constraint of 3D distance can be cast as a 2D constraint which we refer to as the Geometric Correspondence Constraint (GCC). This constraint states that given one pair of correspondences, the two images are partitioned into a family nested of curves such that corresponding points must lie on corresponding curves. This can act as a filter in the RANSAC process with significant savings in computation and with increased robustness and accuracy as demonstrated in experiments using TUM, ICL-NUIM, and RGBD Scene v2 datasets.

Homotopy Continuation (HC) is an effective tool for solving systems of polynomial equations arising from multiview geometry problems in computer vision. Specifically, polynomial systems of camera pose estimation are widely considered as many are used in visual odometry (VO) or simultaneous localization and mapping (SLAM) frameworks. However, existing HC solvers are very slow for many VOs or SLAMs whose efficiency is critical. In addition, in the presence of outliers, the camera pose is typically solved repeatedly under a hypothesis-verification scheme, or RANSAC, for robust estimation, discouraging even more applications from adopting HC as a practical solver. In this paper, we accelerate the state-of-the-art GPU-HC solver progressively by direct parameter homotopy evaluation, improving GPU utilization, pruning homotopy paths, and early termination of RANSAC process, showing around$9.17 \times$,$13.75 \times$,$17.06 \times$, and$17 \times$speedups on NVIDIA V100, A100, H100, and GH200 GPUs, respectively. Additionally, multiple GPUs are also used to demonstrate strong scalability on up to 8 GPUs. We apply the optimized GPU-HC solver on a trifocal pose estimation problem which solves for camera poses using triplet 2D-2D oriented point matches, where one is associated with a 3D point in space. This is particularly useful in recovering VO/SLAM estimation failure when the scene is textureless, as existing recovering methods rely on ample 2D-3D point matches regardless of many 2D-2D pairs present in the images. Experiments show that integrating the accelerated GPU-HC with modern SLAMs substantially increases the success rate of recovering estimation failure without the cost of losing significant runtime.

The semantic reconstruction of a scene relies in part on the curvilinear structure inherent in images. The recovery of curvilinear structure is not only key to the representation of objects via ridges and other object curves but is also critical to the reconstruction from texture-poor images which lack a sufficient number of features. Prior methods advocate for the recovery of curve segments from images and reconstructing these into an organized collection of 3D curve segments often referred to as the 3D curve sketch, which serves as the basis for further reconstruction of curves and surfaces. Observing that the process of edge grouping can lead to fictitious curves or missing veridical groupings, this paper advocates for a reconstruction of curvilinear structure directly from image edges in the form of a 3D edge sketch. The multiview reconstruction of edges faces significant combinatorial challenges which are effectively addressed in this paper. We demonstrate through experiments that the 3D edge sketch recovers a vast majority of the curvilinear structure and is a reliable substrate from which 3D curves can be constructed.

For a generalized (or non-central) camera model, the minimal problem for two views of six points has efficient solvers. However, minimal problems of three views with four points and three views of six lines have not yet been explored and solved, despite the efforts from the computer vision community. This paper develops the formulations of these two minimal problems and shows how state-of-the-art GPU implementations of Homotopy Continuation solver can be used effectively. The proposed methods are evaluated on both synthetic and real datasets, demonstrating that they are fast, accurate and that they improve on structure from motion estimations, when employed in an hypothesis and test setting.

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.