Topics in Tomography

  • Avner Friedman
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 31)


In computerized tomography X-ray transmission measurements are recorded on a computer memory and a mathematical algorithm is applied to produce a numerical description of the tissue density as a function of position within a thin slice of the body. This function is then displayed visually. The X-ray machine projects several hundred parallel pencil beams in the plane of the slice, and the attenuation of each beam is recorded. This procedure is then repeated many times with a small change in the angle at each time. The mathematical problem is to devise an efficient algorithm for computing the density function from the X-ray measurements.


Inverse Fourier Transform Tissue Density Thin Slice Mathematical Algorithm Numerical Description 
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]
    L.A. Shepp and B.F. Logan, The Fourier reconstruction of a head section, IEEE Trans. Nuclear Science, N—S 21 (1974), 21–43.Google Scholar
  2. [2]
    L.A. Shepp and J.B. Kruskal, Computerized tomography: the new medical X-ray technology, Amer. Math. Monthly, 85 (1978), 420–439.CrossRefGoogle Scholar
  3. [3]
    J.A. Reeds and L.A. Shepp, Limited angle reconstruction in tomography via squashing, IEEE Trans. Medical Imaging, MI-16 (1987), 89–97.Google Scholar
  4. [4]
    S. Gutmann, J.H.B. Kemperman, J.A. Reeds and L.A. Shepp, Existence of probability measures with given marginals,to appear in the Annals of Probability.Google Scholar
  5. [5]
    P.C. Fishburn, J.C. Lagarias, J.A. Reeds and L.A. Shepp, Sets uniquely determined by projections on axes I. Continuous case, SIAM J. Appl. Math., 50 (1990), 288–306.Google Scholar
  6. [6]
    R. Gordon, R. Bender and G.T. Herman, Algebraic reconstruction techniques (ART) for three dimensional electron microscopy and X-ray photography, J. Theor. Biol., 29 (1970), 471–481.CrossRefGoogle Scholar
  7. [7]
    G.N. Hounsfield, A method of and apparatus for examination of a body by radiation such as X or gamma radiation,The Patent Office, London (1972), Patent Specification 1283915Google Scholar
  8. [8]
    J. Radon, Ober die Bestimmung von Funktionen durch ihre Integralwerte kings gewissen Mannigfaltigkeiten, Berichte Saechsische Akademie der Wissenschaften, 69 (1917), 267–277.Google Scholar
  9. [9]
    R.N. Bracewell and A.C. Riddle, Inversion of fan-beam scans in radio astronomy, Astro Phys. J., 150 (1967), 427–434.CrossRefGoogle Scholar
  10. [10]
    G.N. Ramachandran and A.V. Lakshinarayanan, Three dimensional reconstruction from radiographs and electron micrographs: application of convolution instead of Fourier transforms, Proc. Nat. Acad. Sci., 68 (1971), 2236–2240.CrossRefGoogle Scholar
  11. [11]
    M. EM—Gal, The shadow transform: An approach to cross-sectional imaging, Stanford Univ. Tech. Report No. 6851–1 (1974).Google Scholar
  12. [12]
    S. Helgason, The Radon transform in Euclidean spaces, compact two point homogeneous spaces and Grassman manifolds, Acta Math., 113 (1965), 153–180.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1990

Authors and Affiliations

  • Avner Friedman
    • 1
  1. 1.Institute for Mathematics and its ApplicationsUniversity of MinnesotaMinneapolisUSA

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