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Projective reconstruction in algebraic vision. (English) Zbl 1448.14062

In recent years, multiview varieties have been extensively studied from the viewpoint of algebraic geometry. A multiview variety \(X\) is the Zariski closure of all images from a sequence of \(n\) “cameras” which are typically projective maps onto low dimensional image spaces (projective planes in the classical scenary of photogrammetry or computer vision). The aim is to use knowledge about \(X\) for the reconstruction of the cameras, up to projective transformation, from measured data on \(X\).
In this paper, the authors prove that the reconstruction is unique unless one uses \(n+1\) cameras onto images spaces of dimension one in which case two solutions exist. This is a known result by R. I. Hartley and F. Schaffalitzky [Lect. Notes Comput. Sci. 3021, 363–375 (2004; Zbl 1098.68775)] but with a new algebro-geometric proof. The authors also prove that the map that sends a camera configuration to the multiview variety is dominant onto an irreducible component of the Hilbert schemes of projective subschemes.

MSC:

14Q65 Geometric aspects of numerical algebraic geometry
14E05 Rational and birational maps
68T45 Machine vision and scene understanding
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
14C05 Parametrization (Chow and Hilbert schemes)

Citations:

Zbl 1098.68775
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Full Text: DOI arXiv

References:

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