Smooth determinantal varieties and critical loci in Multiview Geometry

TL;DR

This paper classifies all smooth critical loci in multiview geometry, showing they are classical projective varieties, which aids in understanding when scene reconstruction from multiple images may fail.

Contribution

It provides a complete classification of smooth critical loci in multiview geometry, linking them to classical projective varieties and enhancing understanding of reconstruction failures.

Findings

Critical loci are determinantal varieties.

All smooth critical loci are classical projective varieties.

Classification aids in understanding scene reconstruction failures.

Abstract

Linear projections from P^k to 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 many enough 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 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.