Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

The ideal of the trifocal variety
HTML articles powered by AMS MathViewer

by Chris Aholt and Luke Oeding PDF
Math. Comp. 83 (2014), 2553-2574 Request permission

Abstract:

Techniques from representation theory, symbolic computational algebra, and numerical algebraic geometry are used to find the minimal generators of the ideal of the trifocal variety. An effective test for determining whether a given tensor is a trifocal tensor is also given.
References
  • M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
  • C. Aholt, B. Sturmfels, and R. Thomas, A Hilbert scheme in computer vision, Canadian Journal of Mathematics (2012), electronic, doi:10.4153/CJM-2012-023-2.
  • Alberto Alzati and Alfonso Tortora, A geometric approach to the trifocal tensor, J. Math. Imaging Vision 38 (2010), no. 3, 159–170. MR 2726517, DOI 10.1007/s10851-010-0216-4
  • D. J. Bates, J. D. Hauenstein, A. J. Sommese, and C. W. Wampler, Bertini: Software for numerical algebraic geometry, Available at http://www. nd.edu/~sommese/bertini, 2010.
  • Winfried Bruns and Udo Vetter, Determinantal rings, Lecture Notes in Mathematics, vol. 1327, Springer-Verlag, Berlin, 1988. MR 953963, DOI 10.1007/BFb0080378
  • David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
  • William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
  • O. Faugeras and B. Mourrain, On the geometry and algebra of the point and line correspondences between $N$ images Proceedings of the Fifth International Conference on Computer Vision, 1995.
  • D. R. Grayson and M. E. Stillman, Macaulay 2, a software system for research in algebraic geometry, 2002.
  • Anders Heyden, Tensorial properties of multiple view constraints, Math. Methods Appl. Sci. 23 (2000), no. 2, 169–202. MR 1738350, DOI 10.1002/(SICI)1099-1476(20000125)23:2<169::AID-MMA110>3.0.CO;2-Y
  • Richard Hartley and Andrew Zisserman, Multiple view geometry in computer vision, 2nd ed., Cambridge University Press, Cambridge, 2003. With a foreword by Olivier Faugeras. MR 2059248
  • J. M. Landsberg, Tensors: geometry and applications, Graduate Studies in Mathematics, vol. 128, American Mathematical Society, Providence, RI, 2012. MR 2865915, DOI 10.1090/gsm/128
  • J. M. Landsberg and L. Manivel, On the ideals of secant varieties of Segre varieties, Found. Comput. Math. 4 (2004), no. 4, 397–422. MR 2097214, DOI 10.1007/s10208-003-0115-9
  • J. M. Landsberg and Jerzy Weyman, On the ideals and singularities of secant varieties of Segre varieties, Bull. Lond. Math. Soc. 39 (2007), no. 4, 685–697. MR 2346950, DOI 10.1112/blms/bdm049
  • Kok Onn Ng, The classification of $(3,3,3)$ trilinear forms, J. Reine Angew. Math. 468 (1995), 49–75. MR 1361786, DOI 10.1515/crll.1995.468.49
  • A. G. Nurmiev, Closures of nilpotent orbits of cubic matrices of order three, Uspekhi Mat. Nauk 55 (2000), no. 2(332), 143–144 (Russian); English transl., Russian Math. Surveys 55 (2000), no. 2, 347–348. MR 1781073, DOI 10.1070/rm2000v055n02ABEH000279
  • A. G. Nurmiev, Orbits and invariants of third-order matrices, Mat. Sb. 191 (2000), no. 5, 101–108 (Russian, with Russian summary); English transl., Sb. Math. 191 (2000), no. 5-6, 717–724. MR 1773770, DOI 10.1070/SM2000v191n05ABEH000478
  • T. Papadopoulo and O. Faugeras, A new characterization of the trifocal tensor, Proceedings of the 5th European Conference on Computer Vision (Freiburg, Germany) (Hans Burkhardt and Bernd Neumann, eds.), Lecture Notes in Computer Science, vol. 1406–1407, Springer–Verlag, June 1998.
  • C. Ressl, Geometry, constraints, and computation of the trifocal tensor, Ph.D. thesis, Universität Bonn, 2003.
  • A.J. Sommese and C.W. Wampler, Numerical solution of polynomial systems arising in engineering and science, World Scientific, Singapore, 2005.
  • R. M. Thrall and J. H. Chanler, Ternary trilinear forms in the field of complex numbers, Duke Math. J. 4 (1938), no. 4, 678–690. MR 1546088, DOI 10.1215/S0012-7094-38-00459-4
  • È. B. Vinberg and A. G. Èlašvili, A classification of the three-vectors of nine-dimensional space, Trudy Sem. Vektor. Tenzor. Anal. 18 (1978), 197–233 (Russian). MR 504529
Similar Articles
Additional Information
  • Chris Aholt
  • Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
  • Email: aholtc@uw.edu
  • Luke Oeding
  • Affiliation: Department of Mathematics, University of California Berkeley, Berkeley, California 94720
  • Address at time of publication: Department of Mathematics and Statistics, 336 Parker Hall, Auburn University, Auburn, Alabama 36849
  • Email: oeding@auburn.edu
  • Received by editor(s): June 22, 2012
  • Received by editor(s) in revised form: January 27, 2013
  • Published electronically: April 17, 2014
  • Additional Notes: The second author was partially supported by NSF RTG Award # DMS-0943745
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 83 (2014), 2553-2574
  • MSC (2010): Primary 13Pxx, 14Qxx; Secondary 15A69, 15A72, 68T45
  • DOI: https://doi.org/10.1090/S0025-5718-2014-02842-1
  • MathSciNet review: 3223346