Numerically Computing Galois Groups of Minimal Problems

TL;DR

This paper explores the intersection of algebra, numerical methods, and computer vision to develop practical approaches for solving parametric algebraic systems, crucial for robust model fitting in vision tasks.

Contribution

It introduces a numerical approach to compute Galois groups of minimal problems, advancing the understanding of their intrinsic difficulty and aiding in solving parametric systems effectively.

Findings

Developed numerical methods for Galois group computation

Measured intrinsic difficulty of solving parametric systems

Improved practical solution strategies for algebraic systems in vision

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.