Accelerating Homotopy Continuation with GPUs: Application to Trifocal Pose Estimation

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.