Families of critical loci in multiview geometry

Linear projections from P^k to P^h model pinhole cameras in the context of Computer Vision or Multiview Geometry. It is well known that, given two sets of n projections P_1,...,P_n and Q_1,...,Q_n, there exist sets of points that have the same images when projected from the two different sets of projection. Such points fill the so–called critical locus for the reconstruction problem for the two sets of projections. In the present paper, we address the problem of describing the critical loci that arise when we keep fixed Q_1,...,Q_n and we allow P_1,...,P_n to vary. In particular, we construct a suitable space that parameterizes the projections P_1,...,P_n, provide an embedding of such space into a suitable Grassmann variety, and construct a map from that space to the Hilbert scheme of closed subschemes in P^k. The subscheme of the Grassmannian corresponding to projections for which the critical locus is the whole P^k is completely characterized, while the fibers of the map above are studied in the case of two projections.