Tomography on the 3D-Torus and Crystals

  • Pablo M. Salzberg
  • Raul Figueroa
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


We exhibit a fast Radon transform on an ambient space over a finite field which furnishes spatial limited-angle models for electron and X-ray tomography. These algorithms have applications in crystallography.


Tilt Angle Binary Tomography Finite Field Ambient Space Binary Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    G. T. Herman, “Two direct methods for reconstructing pictures from their projections: A comparative study,” Computer Graphics and Image Processing 1, 123–144 (1972).CrossRefGoogle Scholar
  2. [2]
    R. A. Gordon, “A tutorial on ART (Algebraic Reconstruction Techniques),” IEEE Trans. Nuclear Science 21 78–93 (1974).CrossRefGoogle Scholar
  3. [3]
    R. A. Brooks and G. Di Chiro, “Theory of image reconstruction in computed tomography,” Radiology 117 561–572 (1975).CrossRefPubMedGoogle Scholar
  4. [4]
    R. A. Brooks and G. Di Chiro, “Principles of computer assisted tomography (CAT) in radiographic and radioisotopic imaging,” Physics in Medicine and Biology 21 689–732 (1976).CrossRefPubMedGoogle Scholar
  5. [5]
    G. T. Herman and R. M. Lewitt, “Overview of image reconstruction from projections,” In G. T. Herman, Image Reconstruction from Projections: Implementation and Applications (Springer-Verlag, Berlin), pp. 1–7, 1979.CrossRefGoogle Scholar
  6. [6]
    D. E. Kuhl and R. Q. Edwards, “Reorganizing data from transverse section scans of the brain using digital processing,” Radiology 91 975–983 (1968).CrossRefPubMedGoogle Scholar
  7. [7]
    M. Hall Jr., Combinatorial Theory, 2 Edition, (John Wiley and Sons, New York), 1986.Google Scholar
  8. [8]
    S. K. Chang and C. K. Chow, “The reconstruction of a three-dimensional object from two orthogonal projections and its applications to cardiac cineangiography,” IEEE Trans. Comput. 22 18–28 (1973).CrossRefGoogle Scholar
  9. [9]
    A. Klug and R. A. Crowther, “Three-dimensional image reconstruction from the viewpoint of information theory,” Nature 238 435–440 (1972).CrossRefGoogle Scholar
  10. [10]
    E. D. Bolker, “The finite Radon transform,” In R. L. Bryant, V. Guillemin, S. Helgason, and R. O. Wells Jr., Integral Geometry, (American Mathematical Society, Providence, RI), pp. 27–50, 1987.CrossRefGoogle Scholar
  11. [11]
    E. L. Grinberg, “The admissibility theorem for the hyperplane transform over a finite field,” J. Combinatorial Th., Series A 53 316–320 (1990).CrossRefGoogle Scholar
  12. [12]
    J. P. S. Kung, “Reconstructing finite Radon transforms,” Nuclear Physics B (Proc. Suppl.) 5A 44–49 (1988).CrossRefGoogle Scholar
  13. [13]
    G. T. Herman, “Reconstruction of binary patterns from a few projections,” In A. Günther, B. Levrat and H. Lipps, International Computing Symposium 1973, (North-Holland Publ. Co., Amsterdam), pp. 371–378,1974.Google Scholar
  14. [14]
    C. Kisielowski, P. Schwander, F. H. Baumann, M. Seibt, Y. Kim, and A. Ourmazd, “An approach to quantitative high-resolution transmission electron microscopy of crystalline materials,” Ultramicroscopy 58 131–155 (1995).CrossRefGoogle Scholar
  15. [15]
    P. Schwander, C. Kisielowski, M. Seibt, F. H. Baumann, Y. Kim, and A. Ourmazd, “Mapping projected potential, interfacial roughness, and composition in general crystalline solids by quantitative transmission electron microscopy,” Phys. Rev. Let. 71 4150–4153 (1993).CrossRefGoogle Scholar
  16. [16]
    P. C. Fishburn, J. C. Lagarias, J. A. Reeds, and L. A. Shepp, “Sets uniquely determined by projections on axes. II. Discrete case,” Disc. Math. 91 149–159, 1991.CrossRefGoogle Scholar
  17. [17]
    R. Aharoni, G. T. Herman, and A. Kuba, “Binary vectors partially determined by linear equation systems,” Disc. Math. 171 1–16, 1997.CrossRefGoogle Scholar
  18. [18]
    A. Kuba, “Reconstruction of unique binary matrices with prescribed elements,” Acta Cybern. 12 57–70 (1995).Google Scholar
  19. [19]
    P. Fishburn, P. Schwander, L. Shepp, and R. J. Vanderbei, “A discrete Radon transform and its approximate inversion via linear programming,” Disc. Appl. Math. 75, 39–61 (1997).CrossRefGoogle Scholar
  20. [20]
    Y. Vardi and D. Lee, “The discrete Radon transform and its approximate inversion via the EM algorithm,” Int. J. Imaging Syst. and Technol. 9, 155–173 (1998).CrossRefGoogle Scholar
  21. [21]
    P. M. Salzberg, P. E. Rivera-Vega, and A. Rodriguez, “Network flow models for binary tomography on lattices,” Int. J. Imaging Syst. and Technol. 9, 147–154 (1998).CrossRefGoogle Scholar
  22. [22]
    P. M. Salzberg, “Tomography in projective spaces: a heuristic for limited-angle reconstructive models,” SIAM J. Matrix Anal. Appl. 9, 393–398 (1988).CrossRefGoogle Scholar
  23. [23]
    P. M. Salzberg, “An application of finite field theory to computerized tomography: A spatial limited-angle model,” In G. L. Mullen and P. Shiue Finite fields, Coding Theory, and Advances in Communications and Computing, (Marcel Dekker, New York), pp. 395–402, 1992.Google Scholar
  24. [24]
    P. M. Salzberg, A. Correa, and R. Cruz, “On a spatial limited-angle model for X-ray computerized tomography,” In Tomography,Impedance Imaging and Integral Geometry, (Amer. Math. Soc., Providence, RI), pp. 25–33, 1994.Google Scholar
  25. [25]
    R. Figueroa and P. M. Salzberg, “Pencil of lines on the 2-D torus,” Ars Combinatoria 37, 235–240 (1994).Google Scholar
  26. [26]
    P. M. Salzberg and R. Figueroa, “Incidence pattern of a pencil of lines in the n-dimensional torus,” Congressus Numeratium 97,197–204 (1993).Google Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Pablo M. Salzberg
    • 1
  • Raul Figueroa
    • 2
  1. 1.Department of Mathematics and Computer SciencesUniversity of Puerto RicoPuerto Rico
  2. 2.Department of Mathematics and Computer SciencesUniversity of Puerto RicoPuerto Rico

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