On the Solvability of Viewing Graphs

Trager, Matthew; Osserman, Brian; Ponce, Jean

doi:10.1007/978-3-030-01270-0_20

Cited by 9 publications

(72 citation statements)

References 24 publications

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“…Fig. 1: Viewing graphs with eight vertices that were left undecided in [16] and that we determined to be solvable.…”

Section: Related Workmentioning

confidence: 99%

“…This analysis was later extended in [19], where it was shown that some solvable graphs can be constructed starting from a triangle and adding vertices of degree two one at a time. More recently, Trager et al [16] provided some necessary conditions and a new sufficient condition for graph solvability. Their results permit to completely characterize graphs with at most seven vertices.…”

Section: Related Workmentioning

confidence: 99%

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

Arrigoni

Fusiello

Rizzi

et al. 2024

IEEE Trans. Pattern Anal. Mach. Intell.

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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.

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“…Fig. 1: Viewing graphs with eight vertices that were left undecided in [16] and that we determined to be solvable.…”

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

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

Arrigoni

Fusiello

Rizzi

et al. 2024

IEEE Trans. Pattern Anal. Mach. Intell.

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“…Heyden and Åström [12] and Trager et al [21] show that when the camera foci are not all on a plane, the bifocals are necessary and sufficient to cut out the multiview variety. There has also been work to further reduce this description by considering the minimal number of bifocals needed ( [12], [23]), though we will not address this question here. In this section, we focus on the ideal-theoretic relationship between the bifocal ideal H 2 A and the multiview ideal M A when the camera foci are noncoplanar.…”

Section: The Bifocal Idealmentioning

confidence: 99%

Ideals of the Multiview Variety

Agarwal

Pryhuber

Thomas

2021

IEEE Trans. Pattern Anal. Mach. Intell.

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The multiview variety of an arrangement of cameras is the Zariski closure of the images of world points in the cameras. The prime vanishing ideal of this complex projective variety is called the multiview ideal. We show that the bifocal and trifocal polynomials from the cameras generate the multiview ideal when the foci are distinct. In the computer vision literature, many sets of (determinantal) polynomials have been proposed to describe the multiview variety. We establish precise algebraic relationships between the multiview ideal and these various ideals. When the camera foci are noncoplanar, we prove that the ideal of bifocal polynomials saturate to give the multiview ideal. Finally, we prove that all the ideals we consider coincide when dehomogenized, to cut out the space of finite images.

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“…A number of papers analyze the solvability of SfM by investigating its corresponding viewing graph, in which each node represents a camera and edges represent available fundamental matrices [16,20,23,29,30]. These approaches, however, assume that the cameras are in general position and hence do not determine which viewing graphs are solvable in (possibly partly) collinear settings.…”

Section: Related Workmentioning

confidence: 99%

Averaging Essential and Fundamental Matrices in Collinear Camera Settings

Geifman¹,

Kasten²,

Galun³

2019

Preprint

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Global methods to Structure from Motion have gained popularity in recent years. A significant drawback of global methods is their sensitivity to collinear camera settings. In this paper, we introduce an analysis and algorithms for averaging bifocal tensors (essential or fundamental matrices) when either subsets or all of the camera centers are collinear. We provide a complete spectral characterization of bifocal tensors in collinear scenarios and further propose two averaging algorithms. The first algorithm uses rank constrained minimization to recover camera matrices in fully collinear settings. The second algorithm enriches the set of possibly mixed collinear and non-collinear cameras with additional, "virtual cameras," which are placed in general position, enabling the application of existing averaging methods to the enriched set of bifocal tensors. Our algorithms are shown to achieve state of the art results on various benchmarks that include autonomous car datasets and unordered image collections in both calibrated and unclibrated settings.

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On the Solvability of Viewing Graphs

Cited by 9 publications

References 24 publications

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

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

Ideals of the Multiview Variety

Averaging Essential and Fundamental Matrices in Collinear Camera Settings

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