Smooth determinantal varieties and critical loci in multiview geometry

Linear projections from $$\mathbb {P}^k$$ to $$\mathbb {P}^h$$ appear in computer vision as models of images of dynamic or segmented scenes. Given multiple projections of the same scene, the identification of sufficiently many correspondences between the images allows, in principle, to reconstruct the position of the projected objects. A critical locus for the reconstruction problem is a variety in $$\mathbb {P}^k$$ containing the set of points for which the reconstruction fails. Critical loci turn out to be determinantal varieties. In this paper we determine and classify all the smooth critical loci, showing that they are classical projective varieties.