Image Reconstruction for Medical Applications

  • David W. Townsend
Part of the EurographicSeminars book series (FOCUS COMPUTER)


The goal of medical image reconstruction is to recover the spatial distribution of a parameter such as the X-ray attenuation coefficient, or radioisotope concentration, inside the body from external measurements, i.e. non-invasive. When the behaviour of the parameter is determined as a function of depth within the body, the resulting image is tomographic, and the procedure is termed tomographic image reconstruction. The essential feature of this procedure is that the image is reconstructed from projections of the underlying distribution. Image reconstruction from projections is a problem that has arisen repeatedly over the past thirty years in a wide variety of different disciplines as diverse as electron microscopy and radioastronomy.


Spatial Domain Reconstruction Algorithm Projection Data Inverse Fourier Transform Projection Direction 
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. Barrett H.H. (1988) The Fundamentals of the Radon Transform. In Proceedings of the NATO ASI on Mathematics and Computer Science in Medical Imaging, Il Ciocco, Italy. Series F, 39. pp. 105–125.Google Scholar
  2. Bates R.H.T. and Peters T.M. (1971) Towards improvements in tomography. N.ZJ. Sci. 14, pp. 883–896.Google Scholar
  3. Berry M.V. and Gibbs D.F. (1970) The interpretation of optical projections. Proc. R. Soc. A, 314, pp. 143–152.CrossRefGoogle Scholar
  4. Bracewell R.N. (1956) Strip integration in radio astronomy. Aust. J. Phys. 9, pp. 198–217.MathSciNetMATHCrossRefGoogle Scholar
  5. Bracewell R.N. (1965) The Fourier Transform and its Applications. New York, McGraw-Hill.MATHGoogle Scholar
  6. Brigham E.O. (1974) The Fast Fourier Transform. Prentice-Hall, Englewood Cliffs, NJ.MATHGoogle Scholar
  7. Brooks R.A. and Di Chiro (1976) Principles of Computerized tomography (CAT) in radiographic and radioisotopic imaging. Phys. Med. Biol. 21, p. 689.CrossRefGoogle Scholar
  8. Censor, Y. (1983) Finite series-expansion reconstruction methods, Proc. IEEE, 71 p. 409.CrossRefGoogle Scholar
  9. Chu G. and Tam K.C. (1977) 3D imaging in the positron camera using Fourier techniques, Phys. Med. Biol. 22, p. 245.CrossRefGoogle Scholar
  10. Clack R., Townsend D.W. and Defrise M. (1989) An algorithm for three-dimensional reconstruction incorporating cross-plane rays, IEEE Trans. Med. Imag. MI-8, p. 32.CrossRefGoogle Scholar
  11. Colsher J.G. (1980) Fully three-dimensional Positron Emission Tomography, Phys. Med. Biol., 25, p. 103.CrossRefGoogle Scholar
  12. Cormack A.M. (1963) Representation of a function by its line integrals, with some radiological applications. J.Appl. Phys. 34, pp. 2722–2727.MATHCrossRefGoogle Scholar
  13. Cormack A.M. (1964) Representation of a function by its line integrals, with some radiological applications. II. J.Appl. Phys. 35, pp. 2908–2913.MATHCrossRefGoogle Scholar
  14. Davison M.E. (1983) The ill-conditioned nature of the limited angle tomography problem, SIAM J. Appl. Math. 43, 428.MathSciNetMATHCrossRefGoogle Scholar
  15. Defrise M., Kuijk S. and Deconinck F. (1987) A new three-dimensional reconstruction method for positron cameras using plane detectors, Phys. Med. Biol. 33 , p. 43.CrossRefGoogle Scholar
  16. Defrise M, Townsend D.W., Clack R. (1989) Three-dimensional image reconstruction from complete projections. Phys. Med. Biol. 34, pp. 573–587.CrossRefGoogle Scholar
  17. Dempster A.P., Laird N.M. and Rubin D.B. (1977) Maximum Likelihood from incomplete data via the Expectation Maximization algorithm, J. Royal Stat. Soc., B39, p. 1.MathSciNetMATHGoogle Scholar
  18. DeRosier D.J. and Klug A. (1968) reconstruction of three-dimensional structures from electron micrographs. Nature 217, pp. 130–134.CrossRefGoogle Scholar
  19. Eggermont, P.P.B., Herman G.T. and Lent, A. (1981) Iterative algorithms for large partitioned linear systems, with applications to image reconstruction, Linear Algebra and its Applications, 40, p. 37.MathSciNetMATHCrossRefGoogle Scholar
  20. Gordon R., Bender R., and Herman G.T. (1970) Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography, J. Theor. Biol. 29 p. 471.CrossRefGoogle Scholar
  21. Hamming R.W. (1983) Digital Filters Prentice Hall, Englewood Cliffs, NJ.Google Scholar
  22. Herman G.T., Lakshminarayanan A.V. and Naparstek A. (1971) Convolution reconstruction techniques for divergent beams, Comp. Biol. Med. 6, p. 259.CrossRefGoogle Scholar
  23. Herman G.T. and Lent A. (1976) Iterative Reconstruction Algorithms. Comput. Biol. Med. 6, pp. 273–294.CrossRefGoogle Scholar
  24. Herman, G.T. (1980) Image Reconstruction from Projections: the Fundamentals of Computerized Tomography. Academic Press, New York.MATHGoogle Scholar
  25. Hounsfield G.N. (1973) Computerized transverse axial scanning tomography: Part I, description of the system. Br. J. Radiol. 46, pp. 1016–1022.CrossRefGoogle Scholar
  26. Joseph, T.M. (1982) An improved algorithm for reprojecting rays through pixel images, IEEE Trans. Med. Imag. MI-1, p. 192.CrossRefGoogle Scholar
  27. Kashyap R.L. and Mittel M.C. (1973) in the First Int. Joint Conf. Pattern Recognition, Washington DC, PP. 286–292.Google Scholar
  28. Kinahan P.E. and Rogers J.G. (1989) Analytic 3D image reconstruction using all detected events, IEEE Trans. Nucl. Sc. NS-36, p. 964.CrossRefGoogle Scholar
  29. Lighthill MJ. (1958) Introduction to Fourier analysis and generalized functions. Cambridge University Press.CrossRefGoogle Scholar
  30. Llacer J. and Veklerov E. (1989) Feasible images and practical stopping rules for iterative algorithms in emission tomography, IEEE Trans. Med. Imag. MI-8, p. 186.CrossRefGoogle Scholar
  31. Muehllehner G and Wetzel R.A. (1971) Section imaging by computer calculation. J. Nucl. Med. 12, pp. 76–84.Google Scholar
  32. Orlov S.S. (1976) Theory of three-dimensional reconstruction. 1. Conditions of a complete set of projections, Soy. Phys. Crystallography 20 p.312.Google Scholar
  33. Pelc NJ. and Chesler D.A. (1979) Utilization of cross-plane rays for three-dimensional reconstruction by filtered back-projection, J. Comput. Assist. Tomogr. 3, p. 385.CrossRefGoogle Scholar
  34. Peters T.M. and Lewitt R.M. (1977) Computed tomography with fan-beam geometry, J. Comput. Assist. Tomogr. 1, p. 429.CrossRefGoogle Scholar
  35. Ra J.B., Lim C.B., Cho Z.H., Hilal S.K., Correll J. (1982) A true three-dimensional reconstruction algorithm for the spherical positron emission tomograph, Phys. Med. Biol. 27, p. 37.CrossRefGoogle Scholar
  36. Radon, J. (1917) Uber die Bestimmung von Funktionen durch ihre Integralwerte langs gewisser Manningsfaltigkeiten, Ber. Verh. Saechs. Akad. Leipzig, Math. Phys. Kl., 69, 262, 1917. Translation by Parks P.C. (1986) in IEEE Trans. Med. Imag. MI-5 p.170.Google Scholar
  37. Ramachandian G.N. and Lakshminarayanan A.V. (1971) Three-dimensional reconstruction from radiographs and electron micrographs: application of convolution instead of Fourier transform, Proc. Nat. Acad. Sc. USA 68 p. 2236.CrossRefGoogle Scholar
  38. Rogers J.G., Harrop R., Kinahan P.E. (1987) The theory of three-dimensional image reconstruction, IEEE Trans. Med. Imaging, MI-6, p. 239.CrossRefGoogle Scholar
  39. Rowland S.W. (1979) Computer implementation of image reconstruction formulas. In Image Reconstruction from Projection : Implementation and Applications. Ed. Herman G.T., Springer-Verlag, Berlin p. 9.CrossRefGoogle Scholar
  40. Rowley P.D. (1969) J. Opt. Soc. Am. 59, pp.1496–1498.CrossRefGoogle Scholar
  41. Schon B., Townsend D. and Clack R. (1983) A general method of 3D filter computation, Phys. Med. Biol. 28, p. 1009.CrossRefGoogle Scholar
  42. Shepp L.A. and Logan B.F. (1974) The Fourier reconstruction of a head section. IEEE Trans. Nucl. Sci. NS-21, pp.21–43.CrossRefGoogle Scholar
  43. Shepp L.A. (1980) Computerized tomography and nuclear magnetic resonance. J. Comput. Assist. Tomogr. 4, pp. 94–107.MathSciNetCrossRefGoogle Scholar
  44. Shepp L.A. and Vardi V. (1982) Maximum Likelihood reconstruction for Emission Computed Tomography, IEEE Trans. Med. Imag., MI-1, p. 113.CrossRefGoogle Scholar
  45. Smith P.R., Peters T.M. and Bates R.H.T. (1973) Image reconstruction from a finite number of projections. J. Phys. A6, pp. 361–382.Google Scholar
  46. Solmon D.C. (1976) The X-ray transform J. Math. Anal. Appl. 56 p. 61.MathSciNetMATHCrossRefGoogle Scholar
  47. Tretiak O.J., Eden M, Simon W. (1969) Internal structure from X-ray images. Proc. 8th Int. Conf. on Med. Biol. Eng., Chicago, Session 12–1.Google Scholar

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© EUROGRAPHICS The European Association for Computer Graphics 1991

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  • David W. Townsend

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