Revisiting Viewing Graph Solvability: an Effective Approach Based on Cycle Consistency

Abstract

In the structure from motion, the viewing graph is a graph where the vertices correspond to cameras (or images) and the edges represent the fundamental matrices. We provide a new formulation and an algorithm for determining whether a viewing graph is solvable, i.e., uniquely determines a set of projective cameras. The known theoretical conditions either do not fully characterize the solvability of all viewing graphs, or are extremely difficult to compute because they involve solving a system of polynomial equations with a large number of unknowns. The main result of this paper is a method to reduce the number of unknowns by exploiting cycle consistency. We advance the understanding of solvability by (i) finishing the classification of all minimal graphs up to 9 nodes, (ii) extending the practical verification of solvability to minimal graphs with up to 90 nodes, (iii) finally answering an open research question by showing that finite solvability is not equivalent to solvability, and (iv) formally drawing the connection with the calibrated case (i.e., parallel rigidity). Finally, we present an experiment on real data that shows that unsolvable graphs may appear in practice.

Compatibility of Fundamental Matrices for Complete Viewing Graphs

“…Solvability. There has been extensive research on the topic of solvability of viewing graphs in computer vision, as evidenced by various studies such as [2,3,15,17,20,25,26]. A viewing graph is considered solvable if, given a generic set of cameras, their fundamental matrices have a unique solution in terms of cameras up to global projective transformation.…”

Compatibility of Fundamental Matrices for Complete Viewing Graphs

“…Solvability. There has been extensive research on the topic of solvability of viewing graphs in computer vision, as evidenced by various studies such as [3,4,17,19,22,27,28]. A viewing graph is considered solvable if, given a generic set of cameras, their fundamental matrices have a unique solution in terms of cameras up to global projective transformation.…”