An Algebraic Geometry Approach to Viewing Graph Solvability

TL;DR

This work reframes viewing graph solvability in uncalibrated structure-from-motion through an Algebraic Geometry lens, introducing a direct polynomial formulation that links cameras to fundamental matrices. A Jacobian-rank test, grounded in the Fiber Dimension Theorem, characterizes finite solvability and yields a practical criterion: a graph is finitely solvable if the Jacobian with respect to camera variables has rank $11|V|-15$ at a generic solution. The authors propose a node-based formulation using the $\Phi$ map, derive explicit derivative matrices, and show how to partition unsolvable graphs into maximal finitely solvable components with a scalable algorithm. Experiments on synthetic and real SfM data demonstrate substantial reductions in problem size and computational effort while preserving solvability outcomes, highlighting strong practical advantages over edge-based approaches.

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.

Figures (5)

• Figure 1: The solvability problem considers the following theoretical question: given a set of fundamental matrices encoded in a graph, how many camera configurations are compliant with such fundamental matrices?
• Figure 2: The connections among different concepts of solvability. The rank condition in Theorem \ref{['teo:theOne']} was called "infinitesimally solvable" in ArrigoniFusielloAl24.
• Figure 3: Examples of maximal components on synthetic viewing graphs, where each component is represented with a different color.
• Figure 4: Number of rows and columns of the matrix used by Arrigoni et al. ArrigoniPajdlaAl23 and the one from our formulation on large-scale SfM datasets WilsonSnavely14CrandallOwensAl11. Observe that the difference in the number of columns even surpasses one order of magnitude.
• Figure 5: Viewing graph of the Tower of London dataset WilsonSnavely14 and maximal components (color-coded). Edges in the largest components are depicted in blue. A zoom is drawn to better visualize the non-solvable part, comprising three edges which resemble the square topology.

Theorems & Definitions (20)

• Definition 1: Solvable framework TragerOssermanAl18
• Definition 2: Solvable graphTragerOssermanAl18
• Definition 3: Finite solvable graphTragerOssermanAl18
• Remark 1
• Remark 2
• Lemma 1: Fiber Dimension Theorem
• Lemma 2: Lemma 2.4 in Chap. 2.6 ofShafarevich13
• Corollary 1
• Proposition 1
• proof