Critical hypersurfaces and instability for reconstruction of scenes in high dimensional projective spaces

In the context of multiple view geometry, images of static scenes are modeled as linear projections from a projective space P^3 to a projective plane P^2 and, similarly, videos or images of suitable dynamic or segmented scenes can be modeled as linear projections from P^k to P^h, with k>h>=2. In those settings, the projective reconstruction of a scene 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 configuration of points and of centers of projections, in the ambient space, where the reconstruction of a scene fails. Critical loci turn out to be suitable algebraic varieties. In this paper we investigate those critical loci which are hypersurfaces in high dimension complex projective spaces, and we determine their equations. Moreover, to give evidence of some practical implications of the existence of these critical loci, we perform a simulated experiment to test the instability phenomena for the reconstruction of a scene, near a critical hypersurface.

Projective Reconstruction in Algebraic Vision

We discuss the geometry of rational maps from a projective space of an arbitrary dimension to the product of projective spaces of lower dimensions induced by linear projections. In particular, we give an algebro-geometric variant of the projective reconstruction theorem by Hartley and Schaffalitzky.

Projective Invariants from Multiple Images: A Direct and Linear Method

The projective reconstruction of 3D structures from 2D images is a central problem in computer vision. Existing methods for this problem are usually nonlinear or indirect. In the previous direct methods, we usually have to solve a system of nonlinear equations. They are very complicated and hard to implement. The previous linear indirect methods are usually imprecise. This paper presents a linear and direct method to derive projective structures of 3D points from their 2D images. Algorithms to compute projective invariants from two images, three images, and four images are given. The method is clear, simple, and easy to implement. For the first time in the literature, we present explicit linear formulas to solve this problem.Mathematicacodes are provided to demonstrate the correctness of the formulas.