GPU-Based Homotopy Continuation for Minimal Problems in Computer Vision

TL;DR

This paper demonstrates that GPU-accelerated Homotopy Continuation significantly speeds up solving complex polynomial systems in computer vision, enabling efficient solutions to problems previously limited by traditional methods.

Contribution

The paper introduces a GPU-based parallelization of Homotopy Continuation, achieving up to 26x speedup and applying it to complex vision problems beyond traditional elimination methods.

Findings

GPU-HC achieves up to 26x speedup on benchmarks.

GPU-HC successfully solves complex vision problems like 4-view triangulation.

Homotopy Continuation can be effectively parallelized on GPU for computer vision.

Abstract

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 26 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.