Critical Configurations for 1-View in Projections from ℙk → ℙ2

Authors:
Marina Bertolini;Cristina Turrini
Affiliations:
Dipartimento di Matematica, Università di Milano, Milano;Dipartimento di Matematica, Università di Milano, Milano
Venue:
Journal of Mathematical Imaging and Vision
Year:
2007

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Abstract

In this paper we describe, from a theoretical point of view, critical configurations for the projective reconstruction of a set of points, for a single view, i.e. for calibration of a camera, in the case of projections from 驴k to 驴2 for k 驴 4. We give first a general result describing these critical loci in 驴k, which, if irreducible, are algebraic varieties of dimension k驴2 and degree 3. If k=4 they can be either a smooth ruled surface or a cone and if k = 5 they can be a smooth three dimensional variety, ruled in planes, or a cone. If k驴 6, the variety is always a cone, the vertex of which has dimension at least k 驴 6. The reducible cases are studied in Appendix A.These results are then applied to determine explicitly the critical loci for the projections from 驴k which arise from the dynamic scenes in 驴3 considered in [13].