The Multidegree of the Multi-Image Variety

Escobar, Laura; Knutson, Allen

doi:10.1007/978-1-4939-7486-3_13

Cited by 10 publications

(13 citation statements)

References 10 publications

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“…These are recorded by [V n ]. After completion of this article, Laura Escobar and Allen Knutson [13] found a proof for the formula (11). Their derivation in [13] rests on methods from representation theory.…”

Section: The Concurrent Lines Varietymentioning

confidence: 99%

Congruences and concurrent lines in multi-view geometry

Ponce

Sturmfels

Trager

2017

Advances in Applied Mathematics

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We present a new framework for multi-view geometry in computer vision. A camera is a mapping between P 3 and a line congruence. This model, which ignores image planes and measurements, is a natural abstraction of traditional pinhole cameras. It includes two-slit cameras, pushbroom cameras, catadioptric cameras, and many more. We study the concurrent lines variety, which consists of n-tuples of lines in P 3 that intersect at a point. Combining its equations with those of various congruences, we derive constraints for corresponding images in multiple views. We also study photographic cameras which use image measurements and are modeled as rational maps from P 3 to P 2 or P 1 × P 1 .

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Section: The Concurrent Lines Varietymentioning

confidence: 99%

Congruences and concurrent lines in multi-view geometry

Ponce

Sturmfels

Trager

2017

Advances in Applied Mathematics

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“…We show now that this is not the case by providing a counterexample in Proposition 3. 37. Antonio Lerario helped with the geometric interpretation of the example as we present it here.…”

Section: Transitive Action On Tangent Spacesmentioning

confidence: 97%

Isotropic and coisotropic subvarieties of Grassmannians

Kohn

Mathews

2021

Advances in Mathematics

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First of all, I want to express my gratitude towards my advisors Bernd Sturmfels and Peter Bürgisser. I am very happy about our open and trustful relationship. Both of you gave me a lot of support, mathematically and personally, so the German term Doktorvater really applies to both of you. Knowing Peter Bürgisser already since the second year of my Bachelor studies in Paderborn, I received a lot of education, advise and encouragement from his side. Teaching together was always fun, both in Paderborn and Berlin. I am thankful that Bernd Sturmfels gave me the chance to pursue a PhD in algebraic geometry, although I had almost no prior knowledge in this area. Your excitement about mathematics gave me inspiration and motivation to keep on learning and exploring, and I really enjoyed writing papers with you.

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“…Congruences play a role in algebraic vision and multi-view geometry, where cameras are modeled as maps from P 3 to congruences [24]. The multidegree of the image of several of those cameras is computed by Escobar and Knutson in [9].…”

mentioning

confidence: 99%

Secants, Bitangents, and Their Congruences

Kohn

Nødland

Tripoli

2017

Fields Institute Communications

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A congruence is a surface in the Grassmannian Gr(1, P 3 ) of lines in projective 3-space. To a space curve C, we associate the Chow hypersurface in Gr(1, P 3 ) consisting of all lines which intersect C. We compute the singular locus of this hypersurface, which contains the congruence of all secants to C. A surface S in P 3 defines the Hurwitz hypersurface in Gr(1, P 3 ) of all lines which are tangent to S. We show that its singular locus has two components for general enough S: the congruence of bitangents and the congruence of inflectional tangents. We give new proofs for the bidegrees of the secant, bitangent and inflectional congruences, using geometric techniques such as duality, polar loci and projections. We also study the singularities of these congruences.

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The Multidegree of the Multi-Image Variety

Abstract: The multi-image variety is a subvariety of Gr(1, P 3 ) n that models taking pictures with n rational cameras. We compute its cohomology class in the cohomology of Gr(1, P 3 ) n , and from there its multidegree as a subvariety of (P 5 ) n under the Plücker embedding.

Cited by 10 publications

References 10 publications

Congruences and concurrent lines in multi-view geometry

Congruences and concurrent lines in multi-view geometry

Isotropic and coisotropic subvarieties of Grassmannians

Secants, Bitangents, and Their Congruences

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