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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
 MSC Classifications: 14N - unknown classification 14N14Q - unknown classification 14Q68 - unknown classification 68