The ideal of the trifocal variety

Aholt, Chris; Oeding, Luke

doi:10.1090/s0025-5718-2014-02842-1

Cited by 40 publications

(72 citation statements)

References 19 publications

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“…In §4 we apply our results to Nurmiev's classification of normal forms. A similar computation for the so-called trifocal variety was carried out by one of us for low degree covariants in [1] (2012). In §5, we test and determine which invariants vanish on which ranks.…”

Section: Introductionmentioning

confidence: 84%

The 3 × 3 × 3 Hyperdeterminant as a Polynomial in the Fundamental Invariants for $${{SL_3(\mathbb{C}) \times SL_3(\mathbb{C}) \times SL_3(\mathbb{C})}}$$ S L 3 ( C ) × S L 3 ( C ) × S L 3 ( C )

Bremner

Oeding

2014

Math.Comput.Sci.

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Section: Introductionmentioning

confidence: 84%

The 3 × 3 × 3 Hyperdeterminant as a Polynomial in the Fundamental Invariants for $${{SL_3(\mathbb{C}) \times SL_3(\mathbb{C}) \times SL_3(\mathbb{C})}}$$ S L 3 ( C ) × S L 3 ( C ) × S L 3 ( C )

Bremner

Oeding

2014

Math.Comput.Sci.

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“…In the classical case of projections from P 3 to P 2 , the rank of the trifocal tensor is known to be 4, (e.g. see [1], [12]), while the rank of the quadrifocal tensor turns out to be 9, [12]. Nothing further is known in general about the ranks of Grassmann tensors.…”

Section: The Rank Of Trifocal Grassmann Tensorsmentioning

confidence: 99%

“…The rank of sequences of tensors converging to tensors associated with degenerate configurations of projection centers is also considered, giving concrete examples of a wide spectrum of phenomena that can happen. the quadrifocal tensor, respectively, and have been studied extensively, see for example [10], [1], [15], [2], [12]. In a more general setting, these tensors are called Grassmann tensors and were introduced by Hartley and Schaffalitzky, [11], as a way to encode information on corresponding subspaces in multiview geometry in higher dimensions.…”

mentioning

confidence: 99%

The Rank of Trifocal Grassmann Tensors

Bertolini¹,

Besana²,

Bini³

et al. 2020

SIAM J. Matrix Anal. Appl.

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Grassmann tensors arise from classical problems of scene reconstruction in computer vision. Trifocal Grassmann tensors, related to three projections from a projective space of dimension k onto view-spaces of varying dimensions are studied in this work. A canonical form for the combined projection matrices is obtained. When the centers of projections satisfy a natural generality assumption, such canonical form gives a closed formula for the rank of trifocal Grassmann tensors. The same approach is also applied to the case of two projections, confirming a previous result obtained with different methods in [6]. The rank of sequences of tensors converging to tensors associated with degenerate configurations of projection centers is also considered, giving concrete examples of a wide spectrum of phenomena that can happen. the quadrifocal tensor, respectively, and have been studied extensively, see for example [10], [1], [15], [2], [12]. In a more general setting, these tensors are called Grassmann tensors and were introduced by Hartley and Schaffalitzky, [11], as a way to encode information on corresponding subspaces in multiview geometry in higher dimensions. Three of the authors have studied critical loci for projective reconstruction from multiple views, [5], [8], and in this setting Grassmann tensors play a fundamental role, [7], [4].The authors' long-term goal is to study properties such as rank, decomposition, degenerations, and identifiability of Grassmann tensors in higher dimensions, and, when feasible, the varieties parameterizing such tensors.The first step was taken in [6], where three of the authors studied the case of two views in higher dimensions, introducing the concept of generalized fundamental matrices as 2-tensors. That first work contained an explicit geometric interpretation of the rational map associated to the generalized fundamental matrix, the computation of the rank of the generalized fundamental matrix with an explicit, closed formula, and the investigation of some properties of the variety of such objects.The next natural step in the authors' program is the study of trifocal Grassmann tensors, i.e. Grassmann tensors arising from three projections from higher dimensional projective spaces onto view-spaces of varying dimensions. A natural genericity assumption, see Assumption 5.1, allows for suitable changes of coordinates in the view spaces and in the ambient space that give rise to a canonical form for the combined projection matrices. Utilizing such canonical form, the rank of trifocal Grassmann tensors is computed with a closed formula, see Theorem 5.1. When Assumption 5.1 is no longer satisfied, the situation becomes quite intricate. A general canonical form for the combined projection matrices can still be obtained, see Section 6. We conclude with a series of examples in which the rank is computed utilizing the canonical form. These examples illustrate the wide spectrum of possible phenomena that can happen with the specialization of the three centers of projection. In particular, we p...

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“…Their characterization, however, is suitable only for a single essential matrix and not for general multiview settings. Finally, [2,11] explore general algebraic properties of multi-view settings.…”

Section: Related Workmentioning

confidence: 99%

Algebraic Characterization of Essential Matrices and Their Averaging in Multiview Settings

Kasten

Geifman

Galun

2019

2019 IEEE/CVF International Conference on Computer Vision (ICCV)

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Essential matrix averaging, i.e., the task of recovering camera locations and orientations in calibrated, multiview settings, is a first step in global approaches to Euclidean structure from motion. A common approach to essential matrix averaging is to separately solve for camera orientations and subsequently for camera positions. This paper presents a novel approach that solves simultaneously for both camera orientations and positions. We offer a complete characterization of the algebraic conditions that enable a unique Euclidean reconstruction of n cameras from a collection of ( n 2 ) essential matrices. We next use these conditions to formulate essential matrix averaging as a constrained optimization problem, allowing us to recover a consistent set of essential matrices given a (possibly partial) set of measured essential matrices computed independently for pairs of images. We finally use the recovered essential matrices to determine the global positions and orientations of the n cameras. We test our method on common SfM datasets, demonstrating high accuracy while maintaining efficiency and robustness, compared to existing methods.

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The ideal of the trifocal variety

Abstract: Abstract. Techniques from representation theory, symbolic computational algebra, and numerical algebraic geometry are used to find the minimal generators of the ideal of the trifocal variety. An effective test for determining whether a given tensor is a trifocal tensor is also given.

Cited by 40 publications

References 19 publications

The 3 × 3 × 3 Hyperdeterminant as a Polynomial in the Fundamental Invariants for $${{SL_3(\mathbb{C}) \times SL_3(\mathbb{C}) \times SL_3(\mathbb{C})}}$$ S L 3 ( C ) × S L 3 ( C ) × S L 3 ( C )

The 3 × 3 × 3 Hyperdeterminant as a Polynomial in the Fundamental Invariants for $${{SL_3(\mathbb{C}) \times SL_3(\mathbb{C}) \times SL_3(\mathbb{C})}}$$ S L 3 ( C ) × S L 3 ( C ) × S L 3 ( C )

The Rank of Trifocal Grassmann Tensors

Algebraic Characterization of Essential Matrices and Their Averaging in Multiview Settings

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