Two Hilbert Schemes in Computer Vision

Lieblich, Max; Meter, Lucas Van

doi:10.1137/18m1200117

Cited by 7 publications

(4 citation statements)

References 6 publications

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“…The algebraic structures underlying these optimization problems are diverse, often drawing from the sums of squares hierarchy as in [129] or synchronization formulations as in [203,22]. We have not touched on many of the deeper tools from algebraic geometry appearing in algebraic vision, such as in [15,14,93,156,167]. Nor have we addressed the role of nonlinear algebra in applications like photometric stereo [241,114], methods for dynamic scenes [239], or the wide variety of alternate camera models encountered in practice (e.g.…”

Section: Algebraic Vision By Timothy Duffmentioning

confidence: 99%

Nonlinear algebra and applications

Breiding¹,

Çelik²,

Duff³

et al. 2023

NACO

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<p style='text-indent:20px;'>We showcase applications of nonlinear algebra in the sciences and engineering. Our review is organized into eight themes: polynomial optimization, partial differential equations, algebraic statistics, integrable systems, configuration spaces of frameworks, biochemical reaction networks, algebraic vision, and tensor decompositions. Conversely, developments on these topics inspire new questions and algorithms for algebraic geometry.</p>

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Section: Algebraic Vision By Timothy Duffmentioning

confidence: 99%

Nonlinear algebra and applications

Breiding¹,

Çelik²,

Duff³

et al. 2023

NACO

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“…A regular locus is given by U ⊂ E such that all E ∈ U have rank 2 and the kernel of E is not spanned by an isotropic vector. A birationally equivalent branched cover was constructed in [60,90], where the authors construct moduli spaces obtained by letting the absolute conic [38] degenerate to a double line. Explicitly, the branched cover X → E is given by…”

Section: Branched Covers and Monodromy Groupsmentioning

confidence: 99%

Galois/monodromy groups for decomposing minimal problems in 3D reconstruction

Timothy¹,

Korotynskiy²,

Pajdla³

et al. 2021

Preprint

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We consider Galois/monodromy groups arising in computer vision applications, with a view towards building more efficient polynomial solvers. The Galois/monodromy group allows us to decide when a given problem decomposes into algebraic subproblems, and whether or not it has any symmetries. Tools from numerical algebraic geometry and computational group theory allow us to apply this framework to classical and novel reconstruction problems. We consider three classical cases-3-point absolute pose, 5-point relative pose, and 4-point homography estimation for calibrated cameras-where the decomposition and symmetries may be naturally understood in terms of the Galois/monodromy group. We then show how our framework can be applied to novel problems from absolute and relative pose estimation. For instance, we discover new symmetries for absolute pose problems involving mixtures of point and line features. We also describe a problem of estimating a pair of calibrated homographies between three images. For this problem of degree 64, we can reduce the degree to 16, the latter better reflecting the intrinsic difficulty of algebraically solving the problem. As a byproduct, we obtain new constraints on compatible homographies, which may be of independent interest.

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“…These varieties appear in computer vision: Each matrix corresponds to a camera, as taking a picture of a geometric object in P 3 corresponds to a linear map from P 3 to P 2 . For further literature, we refer to [4,19].…”

Section: Classification Of the Irreducible Components Of Special Fibementioning

confidence: 99%

Mustafin varieties, moduli spaces and tropical geometry

Hahn

2020

manuscripta math.

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Mustafin varieties are flat degenerations of projective spaces, induced by a set of lattices in a vector space over a non-archimedean field. They were introduced by Mustafin (Math USSR-Sbornik 34(2):187, 1978) in the 70s in order to generalise Mumford's groundbreaking work on the unformisation of curves to higher dimension. These varieties have a rich combinatorial structure as can be seen in pioneering work of Cartwright et al. (Selecta Math 17(4):757-793, 2011). In this paper, we introduce a new approach to Mustafin varieties in terms of images of rational maps, which were studied in Li (IMRN, 2017). Applying tropical intersection theory and tropical convex hull computations, we use this method to give a new combinatorial description of the irreducible components of the special fibers of Mustafin varieties. Finally, we outline a first application of our results in limit linear series theory.

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Two Hilbert Schemes in Computer Vision

Cited by 7 publications

References 6 publications

Nonlinear algebra and applications

Nonlinear algebra and applications

Galois/monodromy groups for decomposing minimal problems in 3D reconstruction

Mustafin varieties, moduli spaces and tropical geometry

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