Smooth determinantal varieties and critical loci in multiview geometry

Linear projections from $$\mathbb {P}^k$$ P k to $$\mathbb {P}^h$$ 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 $$\mathbb {P}^k$$ 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.

Bad projections of the PSD cone

The image of the cone of positive semidefinite matrices under a linear map is a convex cone. Pataki characterized the set of linear maps for which that image is not closed. The Zariski closure of this set is a hypersurface in the Grassmannian. Its components are the coisotropic hypersurfaces of symmetric determinantal varieties. We develop the convex algebraic geometry of such bad projections, with focus on explicit computations.

The minimality of determinantal varieties

The determinantal variety {\Sigma_{pq}} is defined to be the set of all {p\times q} real matrices with {p\geq q} whose ranks are strictly smaller than q. It is proved that {\Sigma_{pq}} is a minimal cone in {\mathbb{R}^{pq}} and all its strata are regular minimal submanifolds.