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A Hilbert Scheme in Computer Vision

Published online by Cambridge University Press:  20 November 2018

Chris Aholt ,
Bernd Sturmfels  and
Rekha Thomas
Chris Aholt
Affiliation:
Mathematics, University of Washington, Seattle, WA 98195, USA, e-mail: aholtc@uw.edu, rrthomas@uw.edu
Bernd Sturmfels
Affiliation:
Mathematics, University of California, Berkeley, CA 94720, USA, e-mail: bernd@math.berkeley.edu
Rekha Thomas
Affiliation:
Mathematics, University of Washington, Seattle, WA 98195, USA, e-mail: aholtc@uw.edu, rrthomas@uw.edu
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Abstract

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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

14N14Q68multigraded Hilbert Schemecomputer visionmonomial idealGroebner basisgeneric initial ideal
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 5 , 01 October 2013 , pp. 961 - 988
Copyright
Copyright © Canadian Mathematical Society 2013

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