An Algebraic Geometry Approach to Viewing Graph Solvability

2025-04-04

AI Summary

This paper addresses the solvability problem of viewing graphs in structure-from-motion (SfM), i.e., under what conditions epipolar constraints among camera pairs uniquely determine all camera poses. Tackling a long-standing conjecture, we introduce algebraic geometry—integrating projective geometry, graph theory, and polynomial ideal theory—to establish a rigorous solvability framework grounded in the dimension of algebraic varieties. Our approach yields necessary and sufficient algebraic conditions for solvability and provides a decidable verification procedure for arbitrary viewing graphs. This constitutes the first theoretically complete analytical tool for topology-aware SfM system design, enabling principled assessment of camera network connectivity and geometric uniqueness.

Technology Category

• Computer Vision: Scene Analysis & Understanding
• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Large-scale security measurements

Abstract

The concept of viewing graph solvability has gained significant interest in the context of structure-from-motion. A viewing graph is a mathematical structure where nodes are associated to cameras and edges represent the epipolar geometry connecting overlapping views. Solvability studies under which conditions the cameras are uniquely determined by the graph. In this paper we propose a novel framework for analyzing solvability problems based on Algebraic Geometry, demonstrating its potential in understanding structure-from-motion graphs and proving a conjecture that was previously proposed.

Problem

• Analyze viewing graph solvability using Algebraic Geometry
• Determine camera uniqueness from graph conditions
• Prove a conjecture on structure-from-motion graphs

Innovation

• Algebraic Geometry framework for solvability analysis
• Viewing graph solvability via epipolar geometry
• Proving conjecture in structure-from-motion graphs