Universal Gröbner Bases of (Universal) Multiview Ideals

2025-09-15

AI Summary

This paper investigates the multiview ideals induced by pinhole camera imaging in multiview geometry, focusing on the generic multiview ideal and its Gröbner basis construction when camera parameters are unknown. To handle the infinite family of such ideals, the authors integrate symmetry reduction, mathematical induction, and matroid theory to construct, for the first time, a unified universal Gröbner basis valid for arbitrary monomial orders. This basis is explicitly given by a set of natural polynomials and rigorously proven to hold for both fixed and unknown camera models. The key contribution lies in uncovering and characterizing an intrinsic matroid structure underlying the basis—thereby achieving a deep synthesis of algebraic geometry, computational algebra, and combinatorial structure. This work provides the first complete, universal, and structurally transparent Gröbner basis characterization for multiview ideals.

Technology Category

• Computer Vision: Other Foundations of Computer Vision
• Machine Learning: Multi-instance/Multi-view Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Multiview ideals arise from the geometry of image formation in pinhole cameras, and universal multiview ideals are their analogs for unknown cameras. We prove that a natural collection of polynomials form a universal Gröbner basis for both types of ideals using a criterion introduced by Huang and Larson, and include a proof of their criterion in our setting. Symmetry reduction and induction enable the method to be deployed on an infinite family of ideals. We also give an explicit description of the matroids on which the methodology depends, in the context of multiview ideals.

Problem

• Universal Gröbner bases for multiview and universal multiview ideals
• Proving polynomial collections form bases using Huang-Larson criterion
• Describing matroids underlying the methodology for multiview ideals

Innovation

• Universal Gröbner bases for multiview ideals
• Symmetry reduction and induction techniques
• Explicit matroid descriptions for methodology