Open AccessCritical hypersurfaces and instability for reconstruction of scenes in high dimensional projective spaces
Author(s) -
Marina Bertolini,
Luca Magri
Publication year - 2020
Publication title -
machine graphics and vision
Language(s) - English
Resource type - Journals
eISSN - 2720-250X
pISSN - 1230-0535
DOI - 10.22630/mgv.2020.29.1.1
Subject(s) - hypersurface , mathematics , projective test , projective space , projective geometry , instability , computer vision , collineation , dimension (graph theory) , context (archaeology) , locus (genetics) , artificial intelligence , pure mathematics , computer science , geometry , algebraic geometry , physics , paleontology , biochemistry , chemistry , mechanics , gene , biology
In the context of multiple view geometry, images of static scenes are modeled as linear projections from a projective space ℙ3 to a projective plane ℙ2 and, similarly, videos or images of suitable dynamic or segmented scenes can be modeled as linear projections from ℙk to ℙh, 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.