Advertisement

Results, Old and New, in Computed Tomography

  • Adel Faridani
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 90)

Abstract

Computed tomography (CT) entails the reconstruction of a function f from line integrals of f. This mathematical problem is encountered in a growing number of diverse settings in medicine, science, and technology, ranging from the famous application in diagnostic radiology to recent research in quantum optics. As a consequence, many aspects of CT have been extensively studied and are now well understood, thus providing an interesting model case for the study of other inverse problems. Other aspects, notably three-dimensional reconstructions, still provide numerous open problems.

Keywords

Inverse Problem Reconstruction Algorithm Local Tomography Line Integral Exterior Problem 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    C. Berenstein and D. Walnut, Local inversion of the Radon transform in even dimensions using wavelets, in [14, pp. 45–69].Google Scholar
  2. [2]
    A. M. Cormack, Representation of a function by its line integrals, with some radiological applications, J. Appl. Phys., 34 (1963), pp. 2722–2727.MATHCrossRefGoogle Scholar
  3. [3]
    A. M. Cormack, Sampling the Radon transform with beams of finite width, Phys. Med. Biol., 23 (1978), pp. 1141–1148.CrossRefGoogle Scholar
  4. [4]
    S. R. Deans, The Radon transform and some of its applications, Wiley, 1983.Google Scholar
  5. [5]
    M. Défrise and R. Clack, A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection, IEEE Trans. Med. Imag., MI-13 (1994), pp. 186–195.CrossRefGoogle Scholar
  6. [6]
    L. Desbat, Efficient sampling on coarse grids in tomography, Inverse Problems, 9 (1993), pp. 251–269.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    A. Faridani, An application of a multidimensional sampling theorem to computed tomography, in [16, pp. 65–80].Google Scholar
  8. [8]
    A. Faridani, Reconstructing from efficiently sampled data in parallel-beam computed tomography, in G.F. Roach (ed.), Inverse problems and imaging, Pitman Research Notes in Mathematics Series, Vol. 245, Longman, 1991, pp. 68–102.Google Scholar
  9. [9]
    A. Faridani, E. L. Ritman, and K. T. Smith, Local tomography, SIAM J. Appl. Math., 52 (1992), pp. 459–484. Examples of local tomography, SIAM J. Appl. Math., 52 (1992), pp. 1193-1198.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    A. Faridani, D.V. Finch, E. L. Ritman, and K. T. Smith, Local tomography II, submitted to SIAM J. Appl. Math.Google Scholar
  11. [11]
    L. A. Feldkamp, L. C. Davis, and J. W. Kress, Practical cone-beam algorithm, J. Opt. Soc. Am. A, 1 (1984), pp. 612–619.CrossRefGoogle Scholar
  12. [12]
    D. V. Finch, Cone beam reconstruction with sources on a curve, SIAM J. Appl. Math., 45 (1985), pp. 665–673.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    I.M. Gelfand and S. G. Gindikin (eds.), Mathematical problems of tomography, Translations of Mathematical Monographs, Vol. 81, Amer. Math. Soc, 1990.Google Scholar
  14. [14]
    S. Gindikin and P. Michor (eds.), 75 Years of Radon Transform, Conference Proceedings and Lecture Notes in Mathematical Physics, Vol. 4, International Press, Boston, 1Google Scholar
  15. [15]
    P. Grangeat, Mathematical framework of cone beam 3D reconstruction via the first derivative of the Radon transform, in [22, pp. 66–97].Google Scholar
  16. [16]
    E. Grinberg and E.T. Quinto (eds.), Integral Geometry and Tomography, Contemporary Mathematics, Vol. 113, Amer. Math. Soc, Providence, R.I., 1990.Google Scholar
  17. [17]
    C. W. Groetsch, Inverse problems in the mathematical sciences, Vieweg, Braunschweig, 1993.Google Scholar
  18. [18]
    S. Gutmann, J. H. B. Kemperman, J. A. Reeds, and L. A. Shepp, Existence of probability measures with given marginals, Annals of Probability, 19 (1991), pp. 1781–1797.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    S. Helgason, The Radon transform, Birkhäuser, 1980.Google Scholar
  20. [20]
    G. T. Herman, Image reconstruction from projections: the fundamentals of computerized tomography, Academic Press, New York, 1980.MATHGoogle Scholar
  21. [21]
    G. T. Herman (ed.), Image reconstruction from projections: implementation and applications, Springer-Verlag, 1979.Google Scholar
  22. [22]
    G.T. Herman, A.K. Louis, and F. Natterer (eds.), Mathematical Methods in Tomography, Lecture Notes in Mathematics, Vol. 1497, Springer-Verlag, 1991.Google Scholar
  23. [23]
    A. C. Kak and M. Slaney, Principles of computerized tomographic imaging, IEEE Press, New York, 1988.MATHGoogle Scholar
  24. [24]
    A.I. Katsevich and A. G. Ramm, A method for finding discontinuities of functions from the tomographic data, in [55, pp. 115–123].Google Scholar
  25. [25]
    A.I. Katsevich and A. G. Ramm, Pseudolocal tomography, SIAM J. Appl. Math. 56 (1996), pp. 167–191. Note added in proof: See also New methods for finding jumps of a function from its local tomographic data, Inverse Probelms 11 (1995), pp. 1005-1023; The Radon Transform and Local Tomography, CRC Press, 1996.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    F. Keinert, Inversion of k-plane transforms and applications in computer tomography, SIAM Review, 31 (1989), pp. 273–298.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    S. Kichenassamy, A. Kumar, P. J. Olver, A. Tannenbaum, T. Yezzi, Conformai curvature flows: from phase transitions to active vision, to appear in Arch. Rat. Mech. Anal..Google Scholar
  28. [28]
    W. Klaverkamp, Tomographische Bildrekonstruktion mit direkten algebraischen Verfahren, Ph.D. thesis, Fachbereich Mathematik der Universität Münster, Münster, Germany, 1991.Google Scholar
  29. [29]
    H. Kruse, Resolution of reconstruction methods in computerized tomography, SIAM J. Sci. Stat. Comput., 10 (1989), pp. 447–474.MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    P. Kuchment, K. Lancaster and L. Mogilevskaya, On local tomography, Inverse Problems, 11 (1995), pp. 571–589.MathSciNetMATHCrossRefGoogle Scholar
  31. [31]
    A. V. Lakshminarayanan, Reconstruction from divergent x-ray data, Suny Tech. Report 32, Comp. Sci. Dept., State University of New York, Buffalo, NY, 1975.Google Scholar
  32. [32]
    J. V. Leahy, K. T. Smith, and D. C. Solmon, Uniqueness, nonuniqueness and inversion in the x-ray and Radon problems, submitted to Proc. Internat. Symp. on Ill-posed Problems, Univ. Delaware Newark, 1979. The proceedings did not appear. Some results of this article have been published in [67].Google Scholar
  33. [33]
    R. M. Lewitt, Reconstruction algorithms: transform methods, Proc. IEEE, 71 (1983), pp. 390–408.CrossRefGoogle Scholar
  34. [34]
    R. M. Lewitt, R. H. T. Bates, and T. M. Peters, Image reconstruction from projections: II: Modified back-projection methods, Optik, 50 (1978), pp. 85–109.Google Scholar
  35. [35]
    B. F. Logan, The uncertainty principle in reconstructing functions from projections, Duke Math. J., 42 (1975), pp. 661–706.MathSciNetMATHCrossRefGoogle Scholar
  36. [36]
    A. K. Louis, Nonuniqueness in inverse Radon problems: the frequency distribution of the ghosts, Math. Z., 185 (1984), pp. 429–440.MathSciNetMATHCrossRefGoogle Scholar
  37. [37]
    A. K. Louis, Orthogonal function series expansions and the null space of the Radon transform, SIAM J. Math. Anal., 15 (1984), pp. 621–633.MathSciNetMATHCrossRefGoogle Scholar
  38. [38]
    A. K. Louis, Incomplete data problems in x-ray computerized tomography I. Singular value decomposition of the limited angle transform, Numer. Math., 48 (1986), pp. 251–262.MathSciNetMATHCrossRefGoogle Scholar
  39. [39]
    A. K. Louis, Inverse und schlecht gestellte Probleme, B.G. Teubner, Stuttgart, 1989.MATHCrossRefGoogle Scholar
  40. [40]
    A. K. Louis, Medical imaging: state of the art and future development, Inverse Problems 8 (1992), pp. 709–738.MathSciNetMATHCrossRefGoogle Scholar
  41. [41]
    A. K. Louis, and P. Maass, Contour reconstruction in 3-D x-ray CT, IEEE Trans. Med. Imag., MI-12 (1993), pp. 764–769.CrossRefGoogle Scholar
  42. [42]
    A. K. Louis and F. Natterer, Mathematical problems in computerized tomography, Proc. IEEE, 71 (1983), pp. 379–389.CrossRefGoogle Scholar
  43. [43]
    W. R. Madych, Summability and approximate reconstruction from Radon transform data, in [16, pp. 189–219].Google Scholar
  44. [44]
    P. Maass, The x-ray transform: singular value decomposition and resolution, Inverse Problems, 3 (1987), pp. 729–741.MathSciNetMATHCrossRefGoogle Scholar
  45. [45]
    P. Maass, The interior Radon transform, SIAM J. Appl. Math., 52 (1992), pp. 710–724.MathSciNetMATHCrossRefGoogle Scholar
  46. [46]
    F. Natterer, The Mathematics of Computerized Tomography, Wiley, 1986.Google Scholar
  47. [47]
    F. Natterer, Sampling in fan-beam tomography, SIAM J. Appl. Math., 53 (1993), pp. 358–380.MathSciNetMATHCrossRefGoogle Scholar
  48. [48]
    F. Natterer, Recent developments in x-ray tomography, in [55, pp. 177–198].Google Scholar
  49. [49]
    T. Olson and J. de Stefano, Wavelet localization of the Radon transform, IEEE Trans. Sig. Proc, 42 (1994), pp. 2055–2067.CrossRefGoogle Scholar
  50. [50]
    R. M. Perry, On reconstructing a function on the exterior of a disc from its Radon transform, J. Math. Anal. Appl., 59 (1977), pp. 324–341.MathSciNetMATHCrossRefGoogle Scholar
  51. [51]
    D. A. Popov, On convergence of a class of algorithms for the inversion of the numerical Radon transform, in [13, pp. 7–65].Google Scholar
  52. [52]
    E. T. Quinto, Singular value decompositions and inversion methods for the exterior Radon transform and a spherical transform, J. Math. Anal. Appl., 95 (1983), pp. 437–448.MathSciNetMATHCrossRefGoogle Scholar
  53. [53]
    E.T. Quinto, Tomographic reconstructions from incomplete data — numerical inversion of the exterior Radon transform, Inverse Problems, 4 (1988), pp. 867–876.MathSciNetMATHCrossRefGoogle Scholar
  54. [54]
    E. T. Quinto, Singularities of the x-ray transform and limited data tomography in R2 and R3, SIAM J. Math. Anal., 24 (1993), pp. 1215–1225.MathSciNetMATHCrossRefGoogle Scholar
  55. [55]
    E.T. Quinto, M. Cheney, and P. Kuchment (eds.), Tomography, Impedance Imaging, and Integral Geometry, Lectures in Applied Mathematics, Vol. 30, Amer. Math. Soc, 1994.Google Scholar
  56. [56]
    A. G. Ramm and A.I. Zaslavsky, Reconstructing singularities of a function given its Radon transform, Math. and Comput. Modelling, 18 (1993), pp. 109–138.MathSciNetMATHCrossRefGoogle Scholar
  57. [57]
    P. A. Rattey and A. G. Lindgren, Sampling the 2-D Radon transform, IEEE Trans. Acoust. Speech Signal Processing, ASSP-29 (1981), pp. 994–1002.Google Scholar
  58. [58]
    M.G. Raymer, M. Beck, and D.F. McAllister, Complex wave-field reconstruction using phase-space tomography, Phys. Rev. Lett., 72 (1994), pp. 1137–1140.MathSciNetMATHCrossRefGoogle Scholar
  59. [59]
    E. L. Ritman, J. H. Dunsmuir, A. Faridani, D. V. Finch, K. T. Smith, and P. J. Thomas, Local reconstruction applied to microtomography. This volume.Google Scholar
  60. [60]
    L. A. Shepp and J. B. Kruskal, Computerized tomography: the new medical x-ray technology, Amer. Math. Monthly, 85 (1978), pp. 420–439.MathSciNetMATHCrossRefGoogle Scholar
  61. [61]
    E. A. Sivers, D. L. Halloway, W. A. Ellingson, and J. Ling, Development and application of local 3-D CT reconstruction software for imaging critical regions in large ceramic turbine rotors, in Rev. Prog. Quant. Nondest. Eval.:, D.O. Thompson and D.E. Chimenti (eds.), Plenum, New York, 1993, pp. 357-364.Google Scholar
  62. [62]
    E. A. Sivers, D. L. Halloway, W. A. Ellingson, Obtaining high-resolution images of ceramics from 3-D x-ray microtomography by region-of-interest reconstruction, Ceramic Eng. Sci. Proc, 14, no. 7-8, (1993), pp. 463–472.CrossRefGoogle Scholar
  63. [63]
    K. T. Smith, D.C. Solmon, and S. L. Wagner, Practical and mathematical aspects of the problem of reconstructing objects from radiographs, Bull. Amer. Math. Soc, 83 (1977), pp. 1227–1270. Addendum in Bull. Amer. Math. Soc, 84 (1978), p. 691.MathSciNetMATHCrossRefGoogle Scholar
  64. [64]
    K. T. Smith and F. Keinert, Mathematical foundations of computed tomography, Appl. Optics 24 (1985), pp. 3950–3957.CrossRefGoogle Scholar
  65. [65]
    D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Measurement of the Wigner distribution and the density matrix of a light mode using optical ho-modyne tomography: application to squeezed states and the vacuum, Phys. Rev. Lett., 70 (1993), pp. 1244–1247.CrossRefGoogle Scholar
  66. [66]
    D. C. Solmon, The x-ray transform, J. Math. Anal. Appl., 56 (1976), pp. 61–83.MathSciNetMATHCrossRefGoogle Scholar
  67. [67]
    D. C. Solmon, Nonuniqueness and the null space of the divergent beam x-ray transform, in [16, pp. 243–249].Google Scholar
  68. [68]
    W. J. T. Spyra, A. Faridani, E. L. Ritman, and K. T. Smith, Computed tomo-itgraphic imaging of the coronary arterial tree — use of local tomography, IEEE Trans. Med. Imag., 9 (1990), pp. 1–4.CrossRefGoogle Scholar
  69. [69]
    H. K. Tuy, An inversion formula for cone beam reconstruction. SIAM J. Appl. Math., 43 (1983), pp. 546–552.MathSciNetCrossRefGoogle Scholar
  70. [70]
    I. Vainberg, I. A. Kazak, and V. P. Kurozaev, Reconstruction of the internal three-dimensional structure of objects based on real-time internal projections, Soviet J. Nondestructive Testing, 17 (1981), pp. 415–423.Google Scholar
  71. [71]
    I. Vainberg, I. A. Kazak, and M. L. Faingoiz, X-ray computerized back projection tomography with filtration by double differentiation. Procedure and information features, Soviet J. Nondestructive Testing, 21 (1985), pp. 106–113.Google Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Adel Faridani
    • 1
  1. 1.Department of MathematicsOregon State UniversityCorvallisUSA

Personalised recommendations