Skip to main content
SpringerLink
Log in

Admissible Complexes for the Projective X-ray Transform over a Finite Field

  • Published:
Discrete & Computational Geometry Aims and scope Submit manuscript

Abstract

We consider the X-ray transform in a projective space over a finite field. It is well known (after Bolker) that this transform is injective. We formulate an analog of Gelfand’s admissibility problem for the Radon transform, which asks for a classification of all minimal sets of lines for which the restricted Radon transform is injective. The solution involves doubly ruled quadric surfaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Bolker, E.D.: The finite Radon transform. In: Integral Geometry (Brunswick 1984), Contemp. Math., vol. 63, pp. 27–50. American Mathematical Society, Providence (1987)

  2. Bolker, E.D., Grinberg, E., Kung, J.P.S.: Admissible complexes for the combinatorial Radon transform. A progress report. In: Integral Geometry and Tomography (Arcata 1989), Contemp. Math., vol. 113, pp. 1–3, American Mathematical Society, Providence (1990)

  3. Gel’fand, I.M., Graev, M.I.: Integral transformations connected with straight line complexes in a complex affine space. Dokl. Akad. Nauk SSSR 138(6), 1266–1269 (1961). (in Russian)

    MathSciNet  MATH  Google Scholar 

  4. Gel’fand, I.M., Graev, M.I., Vilenkin, N.Y.: Generalized Functions. Vol. 5: Integral Geometry and Representation Theory. Academic Press, New York/London (1966)

    MATH  Google Scholar 

  5. Grinberg, E.: The admissibility theorem for the hyperplane transform over a finite field. J. Combin. Theory Ser. A 53(2), 316–320 (1990)

    Article  MathSciNet  Google Scholar 

  6. Grinberg, E.L.: The admissibility theorem for the spatial X-ray transform over the two-element field. In: The Mathematical Legacy of Leon Ehrenpreis. Springer Proc. Math., vol. 16, pp. 111–123. Springer, Milan (2012)

  7. Kirillov, A.A.: On a problem of I. M. Gelfand. Dokl. Akad. Nauk SSSR 137(2), 276–277 (1961). (in Russian)

    MathSciNet  Google Scholar 

  8. Radon, J.: Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Berichte über die Verhandlungen der Königlich-Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse 69, 262–277 (1917)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of New Hampshire, Kingsbury Hall, Durham, NH, 03824, USA

    David V. Feldman

  2. Department of Mathematics, University of Massachusetts, 100 Morrissey Boulevard, Boston, MA, 02125, USA

    Eric L. Grinberg

Authors
  1. David V. Feldman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eric L. Grinberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eric L. Grinberg.

Additional information

Editor in Charge: János Pach

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feldman, D.V., Grinberg, E.L. Admissible Complexes for the Projective X-ray Transform over a Finite Field. Discrete Comput Geom 64, 28–36 (2020). https://doi.org/10.1007/s00454-020-00207-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-020-00207-x

Keywords

Mathematics Subject Classification

Search

Navigation