Linear Solvability in the Viewing Graph

Abstract

Abstract. The Viewing Graph [1] represents several views linked by the corresponding fundamental matrices, estimated pairwise. Given a Viewing Graph, the tuples of consistent camera matrices form a family that we call the Solution Set. This paper provides a theoretical framework that formalizes different properties of the topology, linear solvability and number of solutions of multi-camera systems. We systematically characterize the topology of the Viewing Graph in terms of its solution set by means of the associated algebraic bilinear system. Based on this characterization, we provide conditions about the linearity and the number of solutions and define an inductively constructible set of topologies which admit a unique linear solution. Camera matrices can thus be retrieved efficiently and large viewing graphs can be handled in a recursive fashion. The results apply to problems such as the projective reconstruction from multiple views or the calibration of camera networks.

On the Solvability of Viewing Graphs

“…• We present several criteria for deciding whether or not a viewing graph is solvable. After revisiting some results from [10,15] (Section 3.1), we describe a new necessary condition for solvability that is based on the number of edges and vertices of subgraphs (Theorem 2), as well as a sufficient condition based on "moves" for adding new edges to a graph (Theorem 3).…”

“…In other words, solvable viewing graphs describe sets of fundamental matrices that are generically sufficient to recover a camera configuration. Despite its clear significance, the problem of characterizing which viewing graphs are solvable has not been studied much, and only partial answers are available in the literature (mainly in [10,15]). It is quite easy to produce examples of graphs that are solvable, but it is much more challenging, given a graph, to determine whether it is solvable or not.…”

“…Although correspondences have also been characterized for arbitrary numbers of views [3,9,23], very little theoretical work has been devoted to understanding the geometric constraints imposed on configurations of n > 4 cameras by these tensors, including fundamental matrices [12], which are probably by far the most used in practice. Apart from a few works such as [10,15], understanding how many and which fundamental matrices can be used to recover globally consistent camera parameters is a largely unexplored problem.…”

“…Although they provide some useful necessary conditions (see our Proposition 2 and Example 2), they do not address the problem of solvability in general. In [15], Rudi et al also consider viewing graphs, studying mainly whether a configuration can be recovered from a set of fundamental matrices using a linear system. They also present some "composition rules" for merging solvable graphs into larger ones.…”

GPSfM: Global Projective SFM Using Algebraic Constraints on Multi-View Fundamental Matrices

“…However, in typical settings, only a subset of the n 2 pairwise fundamental matrices can be estimated, and the estimated matrices may be subject at times to significant errors. Moreover, fundamental matrices are defined through a homoge-*Equal contributors neous equation and can thus assume any scale factor, but a consistent setting of these scales is important in multiview settings [20,22] (in analogy to resolving the distance between cameras in a calibrated setting). Consequently, accurate recovery of camera matrices is interesting both from theoretical and practical standpoints.…”