Computed Tomography

  • Ronald Bracewell


When a two-dimensional function f(x, y) is line-integrated in the y-direction or, as we say, is projected onto the x-axis, the resulting one-dimensional function ∫ f(x, y) dy does not contain as much information as the original. But other projections, such as the projection ∫ f(x, y) dy′ onto the x′-axis, where (x′, y′) is a rotated coordinate system, contain different information. From a set of such projections one might hope to be able to reconstruct the original function. The best-known example comes from x-ray computed tomography, where Hounsfield’s brain scanner has had a dramatic effect in radiology and dependent fields such as neurology, but reconstruction of projections was already known in more than one branch of astronomy and has continued to arise in many different contexts. Some indispensable preliminary theory for following the reconstruction techniques is given in the preceding chapter. In this chapter the theory is developed in terms of the x-ray application in order to get the benefit of being able to interpret the equations in physical terms.


Impulse Response Linear Density Line Integral Photographic Plate Radon Transformation 
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Literature cited

  1. E. R. Andrew (1980), “Nuclear magnetic resonance imaging: the multiple sensitive point method,” IEEE Trans. Nuclear Science, vol. NS-27, 1232–1238.CrossRefGoogle Scholar
  2. R. H. T. Bates AND M. J. McDonnell (1986), Image Restoration and Reconstruction, Clarendon Press, Oxford.Google Scholar
  3. R. N. Bracewell (1956), “Strip integration in radio astronomy,” Aust. J. Phys., vol. 9, 198–217.MathSciNetMATHCrossRefGoogle Scholar
  4. R. N. Bracewell (1958), “Restoration in the presence of errors,” Proc. Inst. Radio Engrs., vol. 46, pp. 106–111.Google Scholar
  5. R. N. Bracewell (1977), “Correction for collimator width (restoration) in reconstructive x-ray tomography,” J. Computer Assisted Tomography, vol. 2, 6–15.CrossRefGoogle Scholar
  6. R. N. Bracewell AND A. C. Riddle (1967), “Inversion of fan beam scans in radio astronomy,” Astrophys. J., vol. 150, p. 427–434.CrossRefGoogle Scholar
  7. R. N. Bracewell AND J. A. Roberts (1954), “Aerial smoothing in radio astronomy,” Aust. J. Phys., vol. 7, 615–640.CrossRefGoogle Scholar
  8. R. N. Bracewell AND S. J. Wernecke (1975), “Image reconstruction over a finite field of view,” Journal of the Optical Society of America, vol. 65, 1342–1346.CrossRefGoogle Scholar
  9. R. A. Brooks AND G. DiChiro (1976), “Principles of computer-assisted tomography (CAT) in radiographic and isotropic imaging,” Phys. Med. Biol., vol. 21, 689–732.CrossRefGoogle Scholar
  10. J. F. Claerbout (1992), Earth Soundings Analysis, Blackwell, Boston.Google Scholar
  11. A. Cormack (1963), “Representation of a function by its line integrals, with some radiological applications,” J. Appl. Phys., vol. 34, p. 2722–2727.MATHCrossRefGoogle Scholar
  12. T. Crowther (1970), New Scientist, vol. 47, p. 228.Google Scholar
  13. R. A. Crowther AND A. Klug (1974), “Three-dimensional image reconstruction on an extended field—a fast stable algorithm,” Nature, vol. 251, p. 490–492.CrossRefGoogle Scholar
  14. S. R. Deans (1983), The Radon Transform and Some of Its Applications, John Wiley & Sons, N.Y.MATHGoogle Scholar
  15. D. J. DeRosier AND A. Klug (1968), “Reconstruction of three-dimensional structures from electron micrographs,” Nature, vol. 217, p. 130–134.CrossRefGoogle Scholar
  16. M. Ein-Gal (1974), “The Shadow Transform: An Approach to Cross-sectional Imaging,” Ph.D. thesis, Stanford University.Google Scholar
  17. S. Helgason (1980), “The Radon Transform,” Birkhäuser, Boston, Massachusetts.MATHGoogle Scholar
  18. G. T. Herman AND R.M. Lewitt (1979), “Overview of image reconstruction from projections,” in G. T. Herman ed., Image Reconstruction from Projections, vol. 32, 1–8, Springer-Verlag, N.Y.CrossRefGoogle Scholar
  19. G. Hounsfield (1972), EMI-Scanner, A New Perspective on Brain Disease, EMI Central Research Laboratories, Hayes, Middlesex.Google Scholar
  20. A. C. Kak AND M. Slaney (1988), Principles of Computerized Tomographic Imaging, IEEE, N.Y.MATHGoogle Scholar
  21. A. Klug AND R.A. Crowther (1972), “Three-dimensional image reconstruction from the viewpoint of information theory,” Nature, vol. 238, p. 435–440.CrossRefGoogle Scholar
  22. P. C. Lauterbur (1973), “Image formation by induced local interactions: examples employing nuclear magnetic resonance,” Nature, vol. 242, 190–191.CrossRefGoogle Scholar
  23. P. C. Lauterbur AND C-M Lai (1980), “Zeugmatography by reconstruction from projections,” IEEE Trans. Nuclear Science, vol. NS-27, 1227–1231.CrossRefGoogle Scholar
  24. R. M. Lewitt (1983), “Reconstruction algorithms: transform methods,” Proc. IEEE, vol. 71, pp. 390–408.CrossRefGoogle Scholar
  25. M. J. Lighthill (1958), Introduction to Fourier Analysis and Generalized Functions, Cambridge University Press, England.CrossRefGoogle Scholar
  26. A. Mpacovski (1983), Medical Imaging Systems, Prentice-Hall, Englewood Cliffs, N.J.Google Scholar
  27. H. Na (1974), “Guest editorial,” Intl. J. Imaging Systems Technology, vol. 5, 73–74, Special Issue on Computerized Ionospheric Tomography.CrossRefGoogle Scholar
  28. S. J. Norton (1977), “Theory of Acoustic Imaging,” Ph.D. thesis, Stanford University.Google Scholar
  29. p.A. O’BRien (1953), “The distribution of radiation across the solar disc at meter wavelengths,” Mon. Not. Roy. Astronom. Soc., vol. 113, 597–612.Google Scholar
  30. D. A. Pollen, J. R. Lee, AND J. H. Taylor (1971), “How does the striate cortex begin the reconstruction of the visual world?” Science, vol. 173, 74–77.CrossRefGoogle Scholar
  31. J. Radon (1917), “Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” Berichte über die Verhandlungen der Königlichen Sächsischen Gesellschaft der Wissenschuften zu Leipzig. Mathematisch-Physikalische Klasse, vol. 69, 262–277.Google Scholar
  32. G. N. Ramachandran AND V. A. Lakshminarayanan (1971), “Three-dimensional reconstruction from radiographs and electron micrographs; Applications of convolutions instead of Fourier transforms,” Proc. Nat. Acad. Sci. USA, vol. 68, p. 2236–2240.MathSciNetCrossRefGoogle Scholar
  33. S. W. Rowland (1979), “Computer implementation of image reconstruction formulas,” in G. T. Herman, ed., Topics in Applied Physics, vol. 32, 7–79.Google Scholar
  34. R. C. Spindel AND P. F. Worcester (1990), “Ocean acoustic tomography,” Sci. Amer., vol. 263, pp. 94–99.CrossRefGoogle Scholar
  35. M. K. Stehling, R. Turner, AND P. Mansfield (1991), “Echo-planar imaging: magnetic resonance imaging in a fraction of a second,” Science, vol. 254, 43–50.CrossRefGoogle Scholar
  36. J. H. Taylor (1967), “Two-dimensional brightness distributions of radio sources from lunar occultation observations.” Astrophys. J., vol. 150, p. 421–426.Google Scholar
  37. T. Totsuka AND M. Levoy (1993), “Frequency domain volume rendering,” Proc. SIGGRAPH 93, Anaheim, Calif., August 1993, 271–278.Google Scholar
  38. B. K. Vainshtein (1971), “Synthesis of projecting functions,” Soviet Physics-Doklady, vol. 16, p. 66.Google Scholar
  39. J. G. Verly AND R.N. Bracewell (1979), “Blurring in tomograms made with x—ray beams of finite width,” J. Computer Assisted Tomography, vol. 3, 662–678.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Ronald Bracewell
    • 1
  1. 1.Stanford UniversityStanfordUSA

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