Homotopy Continuation Made Easy: Regression-based Online Simulation of Starting Problem-Solution Pairs

2024-11-06

arXiv.org

AI Summary

Homotopy continuation methods for high-dimensional/high-degree 3D geometric solvers suffer from reliance on full complex-root tracking or offline-trained real-solution classifiers, resulting in low efficiency and poor robustness. Method: This paper proposes a novel paradigm—“initial-solution regression + differentiable online geometric simulation + single-path homotopy backtracking”—that jointly couples a deep regression network with a differentiable online simulator to generate consistent problem-solution pairs as homotopy starting points, eliminating both exhaustive root tracking and offline classifiers. Contribution/Results: The method achieves state-of-the-art success rates and runtime efficiency on CPU, significantly suppressing regression-induced errors. It successfully solves long-standing challenges including generalized camera relocalization and joint relative pose-and-scale estimation. By enabling reliable single-solution recovery, it establishes an efficient and robust homotopy-based framework for solving high-degree polynomial systems in geometric vision.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Computer Vision: Low Level & Physics-based Vision
• Intelligent Robots: State Estimation

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

While automatically generated polynomial elimination templates have sparked great progress in the field of 3D computer vision, there remain many problems for which the degree of the constraints or the number of unknowns leads to intractability. In recent years, homotopy continuation has been introduced as a plausible alternative. However, the method currently depends on expensive parallel tracking of all possible solutions in the complex domain, or a classification network for starting problem-solution pairs trained over a limited set of real-world examples. Our innovation consists of employing a regression network trained in simulation to directly predict a solution from input correspondences, followed by an online simulator that invents a consistent problem-solution pair. Subsequently, homotopy continuation is applied to track that single solution back to the original problem. We apply this elegant combination to generalized camera resectioning, and also introduce a new solution to the challenging generalized relative pose and scale problem. As demonstrated, the proposed method successfully compensates the raw error committed by the regressor alone, and leads to state-of-the-art efficiency and success rates while running on CPU resources, only.

Problem

• Overcoming intractability in 3D vision due to high-degree constraints
• Reducing dependency on expensive parallel tracking in homotopy continuation
• Improving efficiency in generalized camera resectioning and pose estimation

Innovation

• Predict rough initial solution for homotopy
• Generate problem via online simulator
• Track single solution to original problem