The Bordiga surface as critical locus for 3-view reconstructions

In Computer Vision, images of dynamic or segmented scenes are modeled as linear projections from Pk to P2. 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 Pk containing the objects for which the reconstruction fails. In this paper, we deal with projections both of points from P4 to P2 and of lines from P3 to P2. 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 Hartley and Schaffalitzky (2004) 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)⊂P5. This last surface is biregular to a Bordiga surface (Verra, 1988). 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.