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A Hilbert Scheme in Computer Vision

  Published:2012-07-19
 Printed: Oct 2013
  • Chris Aholt,
    Mathematics, University of Washington, Seattle, WA 98195, USA
  • Bernd Sturmfels,
    Mathematics, University of California, Berkeley, CA 94720, USA
  • Rekha Thomas,
    Mathematics, University of Washington, Seattle, WA 98195, USA
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Abstract

Multiview geometry is the study of two-dimensional images of three-dimensional scenes, a foundational subject in computer vision. We determine a universal Gröbner basis for the multiview ideal of $n$ generic cameras. As the cameras move, the multiview varieties vary in a family of dimension $11n-15$. This family is the distinguished component of a multigraded Hilbert scheme with a unique Borel-fixed point. We present a combinatorial study of ideals lying on that Hilbert scheme.
Keywords: multigraded Hilbert Scheme, computer vision, monomial ideal, Groebner basis, generic initial ideal multigraded Hilbert Scheme, computer vision, monomial ideal, Groebner basis, generic initial ideal
MSC Classifications: 14N, 14Q, 68 show english descriptions unknown classification 14N
unknown classification 14Q
unknown classification 68
14N - unknown classification 14N
14Q - unknown classification 14Q
68 - unknown classification 68
 

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