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

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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 ℙk to ℙ2 for k ≥ 4. We give first a general result describing these critical loci in ℙk, which, if irreducible, are algebraic varieties of dimension k−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≥ 6, the variety is always a cone, the vertex of which has dimension at least k − 6. The reducible cases are studied in Appendix A.

These results are then applied to determine explicitly the critical loci for the projections from ℙk which arise from the dynamic scenes in ℙ3 considered in [13].

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Milano, Milano

    Marina Bertolini & Cristina Turrini

Authors
  1. Marina Bertolini
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  2. Cristina Turrini
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Correspondence to Marina Bertolini.

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Marina Bertolini is currently Associate Professor of Geometry at the Department of Mathematics at the Università degli Studi di Milano, Italy. Her main field of research is Complex Projective Algebraic Geometry, with particular interest for the classification of projective varieties and for the geometry of Grassmann varieties. On these topics M. Bertolini has published more than twenty reviewed papers on national and international journals. She has been for some years now interested also in applications of Algebraic Geometry to Computer Vision problems.

Cristina Turrini is Associate Professor of Geometry at the Department of Mathematics of Università degli Studi di Milano, Italy. Her main research interest is Complex Projective Algebraic Geometry: subvarieties of Grassmannians, special varieties, automorphisms, classification. In the last two years she has started to work on applications of Algebraic Geometry to problems of Computer Vision. She is author or co-author of about thirty reviewed papers. She is also involved in popularization of Mathematics, and on this subject she is co-editor of some books.

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Bertolini, M., Turrini, C. Critical Configurations for 1-View in Projections from ℙk → ℙ2 . J Math Imaging Vis 27, 277–287 (2007). https://doi.org/10.1007/s10851-007-0649-6

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