The Bordiga surface as critical locus for 3-view reconstructions
| Autor: | Roberto Notari, Christina Turrini, Marina Bertolini |
|---|---|
| Přispěvatelé: | Department of Mathematics 'Federigo Enriques', Università degli Studi di Milano [Milano] (UNIMI), Dipartimento di Matematica, 'Francesco Brioschi', Politecnico di Milano [Milan] (POLIMI) |
| Rok vydání: | 2019 |
| Předmět: |
Algebra and Number Theory
Multiview Geometry 010102 general mathematics Bordiga surface 010103 numerical & computational mathematics 01 natural sciences Combinatorics Computational Mathematics Critical configurations or loci Reconstruction problem Hilbert scheme Grassmannian Projective reconstruction in Computer Vision Bordiga surface Line congruences in Grassmannians Projective reconstruction in Computer Vision Multiview geometry Critical configurations or loci Line congruences in Grassmannians [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] 0101 mathematics Locus (mathematics) Irreducible component Mathematics |
| Zdroj: | MEGA 2017-International Conference on Effective Methods in Algebraic Geometry MEGA 2017-International Conference on Effective Methods in Algebraic Geometry, Jun 2017, Nice, France |
| ISSN: | 0747-7171 |
| DOI: | 10.1016/j.jsc.2018.06.014 |
| Popis: | International audience; In Computer Vision, images of dynamic or segmented scenes are modeled as linear projections from P k to P 2. The reconstruction problem consists in recovering the position of the projected objects and the projections themselves from their images, after identifying many enough correspondences between the images. A critical locus for the reconstruction problem is a variety in P k containing the objects for which the reconstruction fails. In this paper, we deal with projections both of points from P 4 → P 2 and of lines from P 3 → P 2. In both cases, we consider 3 projections, minimal number for a uniquely determined reconstruction. In the case of projections of points, we declinate the Grassmann tensors introduced in [21] in our context, and we use them to compute the equations of the critical locus. Then, given the ideal that defines this locus, we prove that, in the general case, it defines a Bordiga surface, or a scheme in the same irreducible component of the associated Hilbert scheme. Furthermore, we prove that every Bordiga surface is actually the critical locus for the reconstruction for suitable projections. In the case of projections of lines, we compute the defining ideal of the critical locus, that is the union of 3 α–planes and a line congruence of bi–degree (3, 6) and sectional genus 5 in the Grassmannian G(1, 3) ⊂ P 5. This last surface is biregular to a Bordiga surface [40]. We use this fact to link the two reconstruction problems by showing how to compute the projections of one of the two settings, given the projections of the other one. The link is effective, in the sense that we describe an algorithm to compute the projection matrices. |
| Databáze: | OpenAIRE |
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