The Bordiga surface as critical locus for 3-view reconstructions

Abstract

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.

Smooth determinantal varieties and critical loci in multiview geometry

“…The Bordiga surface has been considered from the point of view of critical loci in [8]. In that paper, the authors proved that the critical locus for a couple of three projections from ℙ 4 to ℙ 2 is in the irreducible component of the Hilbert scheme containing the Bordiga surface as general element ( [8], Proposition 5.1), and conversely, that every Bordiga surface X is actually critical for suitable couples of three projections ( [8], Theorem 5.1).…”

“…Later, in [2,[7][8] the study of the ideal of critical loci has been formalized making use of the so-called Grassmann tensor introduced in [18]. A seminal case of this approach has been considered in [2], where the authors computed the equations of the critical locus for three projections from ℙ 2 to ℙ 1 .…”

“…A seminal case of this approach has been considered in [2], where the authors computed the equations of the critical locus for three projections from ℙ 2 to ℙ 1 . In [6,8] the case of three projections from ℙ 4 to ℙ 2 is studied in detail. When the projections are general enough, critical loci are shown to be Bordiga surfaces, and conversely, every Bordiga surface is shown to be critical for suitable choices of three projections.…”

Critical hypersurfaces and instability for reconstruction of scenes in high dimensional projective spaces

“…Critical loci were shown to be special determinantal varieties, and particular attention was given to the case of P 4 in which a Bordiga surface was obtained as essential component of the critical locus. This case has been further investigated in a fully projective context in [5] and in [1]. Finally, critical loci for multiple views, i.e., for projections from P k to P 2 , are extensively considered in [3], where the varieties arising as critical loci turns out to be hypersurfaces of degree r in P 2r−1 or varieties of codimension 2 and degree (r+2)(r+1) 2 in P 2r .…”

The Rank of Trifocal Grassmann Tensors

“…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.…”