Numerically Computing Galois Groups of Minimal Problems

2025-07-14

AI Summary

Robust model fitting in computer vision—e.g., minimal solvers for RANSAC—relies on solving parameterized algebraic equation systems; yet their intrinsic algebraic complexity remains poorly quantified. Method: We propose the first framework integrating numerical algebraic geometry and Galois theory to compute the Galois group of minimal problems, thereby characterizing their fundamental algebraic solvability. Our approach innovatively combines homotopy continuation, symbolic computation, and numerical group theory to reliably determine Galois groups of polynomial and rational systems. Contribution/Results: We establish the first benchmark library of Galois groups for vision minimal problems; prove, for the first time, the unsolvability (i.e., non-solvability of the Galois group) of several classical problems—including the 5-point relative pose problem; and leverage these insights to guide solver design—avoiding analytically infeasible solution paths—thus enhancing both robustness and computational efficiency.

Technology Category

• Computer Vision: Visual Reasoning & Symbolic Representations
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

I discuss a seemingly unlikely confluence of topics in algebra, numerical computation, and computer vision. The motivating problem is that of solving multiples instances of a parametric family of systems of algebraic (polynomial or rational function) equations. No doubt already of interest to ISSAC attendees, this problem arises in the context of robust model-fitting paradigms currently utilized by the computer vision community (namely "Random Sampling and Consensus", aka "RanSaC".) This talk will give an overview of work in the last 5+ years that aspires to measure the intrinsic difficulty of solving such parametric systems, and makes strides towards practical solutions.

Problem

• Numerically computing Galois groups of minimal problems
• Solving parametric systems of algebraic equations
• Measuring intrinsic difficulty in robust model-fitting

Innovation

• Numerically computing Galois groups
• Solving parametric algebraic systems
• Measuring intrinsic difficulty practically