• Gabor T. Herman
Living reference work entry


We define tomography as the process of producing an image of a distribution (of some physical property) from estimates of its line integrals along a finite number of lines of known locations. We touch upon the computational and mathematical procedures underlying the data collection, image reconstruction, and image display in the practice of tomography. The emphasis is on reconstruction methods, especially the so-called series expansion reconstruction algorithms. We illustrate the use of tomography (including three-dimensional displays based on reconstructions) both in electron microscopy and in X-ray computerized tomography (CT), but concentrate on the latter. This is followed by a classification and discussion of reconstruction algorithms. In particular, we discuss how to evaluate and compare the practical efficacy of such algorithms.


Reconstruction Algorithm Projection Data Helical Computerize Tomography Source Position Line Integral 
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.


  1. 1.
    Artzy, E., Frieder, G., Herman, G.T.: The theory, design, implementation and evaluation of a three-dimensional surface detection algorithm. Comput. Graph. Image Proc. 15, 1–24 (1981)CrossRefGoogle Scholar
  2. 2.
    Banhart, J.: Advanced Tomographic Methods in Materials Research and Engineering. Oxford University Press, Oxford (2008)CrossRefGoogle Scholar
  3. 3.
    Bracewell, R.N.: Strip integration in radio astronomy. Aust. J. Phys. 9, 198–217 (1956)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Browne, J.A., De Pierro, A.R.: A row-action alternative to the EM algorithm for maximizing likelihood in emission tomography. IEEE Trans. Med. Imaging 15, 687–699 (1996)CrossRefGoogle Scholar
  5. 5.
    Censor, Y., Altschuler, M.D., Powlis, W.D.: On the use of Cimmino’s simultaneous projections method for computing a solution of the inverse problem in radiation therapy treatment planning. Inverse Prob. 4, 607–623 (1988)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Censor, Y., Chen, W., Combettes, P.L., Davidi, R., Herman G.T.: On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints. Comput. Optim. Appl. 51, 1065–1088 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Censor, Y., Zenios, S.A.: Parallel Optimization: Theory, Algorithms and Applications. Oxford University Press, New York (1998)Google Scholar
  8. 8.
    Chen, L.S., Herman, G.T., Reynolds, R.A., Udupa, J.K.: Surface shading in the cuberille environment (erratum appeared in 6(2):67–69, 1986). IEEE Comput. Graph. Appl. 5(12), 33–43 (1985)CrossRefzbMATHGoogle Scholar
  9. 9.
    Cormack, A.M.: Representation of a function by its line integrals, with some radiological applications. J. Appl. Phys. 34, 2722–2727 (1963)CrossRefzbMATHGoogle Scholar
  10. 10.
    Crawford, C.R., King, K.F.: Computed-tomography scanning with simultaneous patient motion. Med. Phys. 17, 967–982 (1990)CrossRefGoogle Scholar
  11. 11.
    Crowther, R.A., DeRosier, D.J., Klug, A.: The reconstruction of a three-dimensional structure from projections and its application to electron microscopy. Proc. R. Soc. Lond. Ser.-A A317, 319–340 (1970)CrossRefGoogle Scholar
  12. 12.
    Davidi, R., Herman, G.T., Klukowska, J.: SNARK09: A Programming System for the Reconstruction of 2D Images from 1D Projections (2009).
  13. 13.
    DeRosier, D.J., Klug, A.: Reconstruction of three-dimensional structures from electron micrographs. Nature 217, 130–134 (1968)CrossRefGoogle Scholar
  14. 14.
    Edholm, P.R., Herman, G.T.: Linograms in image reconstruction from projections. IEEE Trans. Med. Imaging 6, 301–307 (1987)CrossRefGoogle Scholar
  15. 15.
    Edholm, P., Herman, G.T., Roberts, D.A.: Image reconstruction from linograms: implementation and evaluation. IEEE Trans. Med. Imaging 7, 239–246 (1988)CrossRefGoogle Scholar
  16. 16.
    Eggermont, P.P.B., Herman, G.T., Lent, A.: Iterative algorithms for large partitioned linear systems, with applications to image reconstruction. Linear Algebra Appl. 40, 37–67 (1981)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Epstein, C.S.: Introduction to the Mathematics of Medical Imaging. 2nd edn. SIAM, Philadelphia (2007)CrossRefGoogle Scholar
  18. 18.
    Frank, J.: Electron Tomography: Methods for Three-Dimensional Visualization of Structures in the Cell, 2nd edn. Springer, New York (2006)CrossRefGoogle Scholar
  19. 19.
    Frank, J.: Three-Dimensional Electron Microscopy of Macromolecular Assemblies: Visualization of Biological Molecules in Their Native State. Oxford University Press, Oxford (2006)CrossRefGoogle Scholar
  20. 20.
    Gordon, R., Bender, R., Herman, G.T.: Algebraic Reconstruction Techniques (ART) for three-dimensional electron microscopy and X-ray photography. J. Theory Biol. 29, 471–481 (1970)CrossRefGoogle Scholar
  21. 21.
    Hanson, K.M.: Method of evaluating image-recovery algorithms based on task performance. J. Opt. Soc. Am. A 7, 1294–1304 (1990)CrossRefGoogle Scholar
  22. 22.
    Herman, G.T.: Advanced principles of reconstructing algorithms. In: Newton, T.H., Potts, D.G. (eds.) Radiology of Skull and Brain. Technical Aspects of Computed Tomography, vol. 5, pp. 3888–3903. C.V. Mosby Company, St. Louis (1981)Google Scholar
  23. 23.
    Herman, G.T.: Fundamentals of Computerized Tomography: Image Reconstruction from Projections, 2nd edn. Springer, Berlin (2009)CrossRefGoogle Scholar
  24. 24.
    Herman, G.T., Kuba, A.: Advances in Discrete Tomography and Its Applications. Birkhäuser, Boston (2007)CrossRefzbMATHGoogle Scholar
  25. 25.
    Herman, G.T., Lent, A.: Iterative reconstruction algorithms. Comput. Biol. Med. 6, 273–294 (1976)CrossRefGoogle Scholar
  26. 26.
    Herman, G.T., Liu, H.K.: Three-dimensional display of human organs from computed tomograms. Comput. Graph. Image Proc. 9, 1–21 (1979)CrossRefGoogle Scholar
  27. 27.
    Herman, G.T., Meyer, L.B.: Algebraic reconstruction techniques can be made computationally efficient. IEEE Trans. Med. Imaging 12, 600–609 (1993)CrossRefGoogle Scholar
  28. 28.
    Herman, G.T., Naparstek, A.: Fast image reconstruction based on a Radon inversion formula appropriate for rapidly collected data. SIAM J. Appl. Math. 33, 511–533 (1977)CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Herman, G.T., Tuy, H.K., Langenberg, K.J., Sabatier, P.C.: Basic Methods of Tomography and Inverse Problems. Institute of Physics Publishing, Bristol (1988)Google Scholar
  30. 30.
    Herman, G.T., Garduño, E., Davidi, R., Censor, Y.: Superiorization: an optimization heuristic for medical physics. Med. Phys. 39, 5532–5546 (2012)CrossRefGoogle Scholar
  31. 31.
    Hounsfield, G.N.: Computerized transverse axial scanning tomography: Part I, description of the system. Br. J. Radiol. 46, 1016–1022 (1973)CrossRefGoogle Scholar
  32. 32.
    Hudson, H.M., Larkin, R.S.: Accelerated image reconstruction using ordered subsets of projection data. IEEE Trans. Med. Imaging 13, 601–609 (1994)CrossRefGoogle Scholar
  33. 33.
    Kalender, W.A.: Computed Tomography: Fundamentals, System Technology, Image Quality, Applications, 2nd edn. Wiley-VCH (2006)Google Scholar
  34. 34.
    Kalender, W.A., Seissler, W., Klotz, E., Vock, P.: Spiral volumetric CT with single-breath-hold technique, continuous transport, and continuous scanner rotation. Radiology 176, 181–183 (1990)CrossRefGoogle Scholar
  35. 35.
    Katsevich, A.: Theoretically exact filtered backprojection-type inversion algorithm for spiral CT. SIAM J. Appl. Math. 62, 2012–2026 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    Kinahan, P.E., Matej, S., Karp, J.P., Herman, G.T., Lewitt, R.M.: Comparison of transform and iterative reconstruction techniques for a volume-imaging PET scanner with a large axial acceptance angle. IEEE Trans. Nucl. Sci. 42, 2181–2287 (1995)CrossRefGoogle Scholar
  37. 37.
    Klukowska, J., Davidi, R., Herman, G.T.: SNARK09 – a software package for the reconstruction of 2D images from 1D projections. Comput. Methods Programs Biomed. 110, 424–440 (2013)CrossRefGoogle Scholar
  38. 38.
    Lauterbur, P.C.: Medical imaging by nuclear magnetic resonance zeugmatography. IEEE Trans. Nucl. Sci. 26, 2808–2811 (1979)CrossRefGoogle Scholar
  39. 39.
    Levitan, E., Herman, G.T.: A maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography. IEEE Trans. Med. Imaging 6, 185–192 (1987)CrossRefGoogle Scholar
  40. 40.
    Lewitt, R.M.: Multidimensional digital image representation using generalized Kaiser–Bessel window functions. J. Opt. Soc. Am. A 7, 1834–1846 (1990)CrossRefGoogle Scholar
  41. 41.
    Lewitt, R.M.: Alternatives to voxels for image representation in iterative reconstruction algorithms. Phys. Med. Biol. 37, 705–716 (1992)CrossRefGoogle Scholar
  42. 42.
    Lorensen, W., Cline, H.: Marching cubes: a high-resolution 3D surface reconstruction algorithm. Comput. Graph. 21(4), 163–169 (1987)CrossRefGoogle Scholar
  43. 43.
    Maki, D.D., Birnbaum, B.A., Chakraborty, D.P., Jacobs, J.E., Carvalho, B.M., Herman, G.T.: Renal cyst pseudo-enhancement: beam hardening effects on CT numbers. Radiology 213, 468–472 (1999)CrossRefGoogle Scholar
  44. 44.
    Marabini, R., Rietzel, E., Schroeder, R., Herman, G.T., Carazo, J.M.: Three-dimensional reconstruction from reduced sets of very noisy images acquired following a single-axis tilt schema: application of a new three-dimensional reconstruction algorithm and objective comparison with weighted backprojection. J. Struct. Biol. 120, 363–371 (1997)CrossRefGoogle Scholar
  45. 45.
    Marabini, R., Herman, G.T., Carazo, J.-M.: 3D reconstruction in electron microscopy using ART with smooth spherically symmetric volume elements (blobs). Ultramicroscopy 72, 53–65 (1998)CrossRefGoogle Scholar
  46. 46.
    Matej, S., Lewitt, R.M.: Practical consideration for 3D image-reconstruction using spherically-symmetrical volume elements. IEEE Trans. Med. Imaging 15, 68–78 (1996)CrossRefGoogle Scholar
  47. 47.
    Matej, S., Herman, G.T., Narayan, T.K., Furuie, S.S., Lewitt, R.M., Kinahan, P.E.: Evaluation of task-oriented performance of several fully 3D PET reconstruction algorithms. Phys. Med. Biol. 39, 355–367 (1994)CrossRefGoogle Scholar
  48. 48.
    Matej, S., Furuie, S.S., Herman, G.T.: Relevance of statistically significant differences between reconstruction algorithms. IEEE Trans. Image Proc. 5, 554–556 (1996)CrossRefGoogle Scholar
  49. 49.
    Narayan, T.K., Herman, G.T.: Prediction of human observer performance by numerical observers: an experimental study. J. Opt. Soc. Am. A 16, 679–693 (1999)CrossRefGoogle Scholar
  50. 50.
    Natterer, F., Wübbeling, F.: Mathematical Methods in Image Reconstruction. SIAM, Philadelphia (2001)CrossRefzbMATHGoogle Scholar
  51. 51.
    Poulsen, H.F.: Three-Dimensional X-Ray Diffraction Microscopy: Mapping Polycrystals and Their Dynamics. Springer, Berlin (2004)CrossRefGoogle Scholar
  52. 52.
    Radon, J.: Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Ber. Verh. Sächs. Akad. Wiss., Leipzig, Math. Phys. Kl. 69, 262–277 (1917)Google Scholar
  53. 53.
    Ramachandran, G.N., Lakshminarayanan, A.V.: Three-dimensional reconstruction from radiographs and electron micrographs: application of convolutions instead of Fourier transforms. Proc. Natl. Acad. Sci. USA 68, 2236–2240 (1971)CrossRefMathSciNetGoogle Scholar
  54. 54.
    Scheres, S.H.W., Gao, H., Valle, M., Herman, G.T., Eggermont, P.P.B., Frank, J., Carazo, J.-M.: Disentangling conformational states of macromolecules in 3D-EM through likelihood optimization. Nat. Methods 4, 27–29 (2007)CrossRefGoogle Scholar
  55. 55.
    Scheres, S.H.W., Nuñez-Ramirez, R., Sorzano, C.O.S., Carazo, J.M., Marabini, R.: Image processing for electron microscopy single-particle analysis using XMIPP. Nat. Protoc. 3, 977–990 (2008)CrossRefGoogle Scholar
  56. 56.
    Shepp, L.A., Logan, B.F.: The Fourier reconstruction of a head section. IEEE Trans. Nucl. Sci. 21, 21–43 (1974)CrossRefGoogle Scholar
  57. 57.
    Shepp, L.A., Vardi, Y.: Maximum likelihood reconstruction for emission tomography. IEEE Trans. Med. Imaging 1, 113–122 (1982)CrossRefGoogle Scholar
  58. 58.
    Sorzano, C.O.S., Marabini, R., Boisset, N., Rietzel, E., Schröder, R., Herman, G.T., Carazo, J.M.: The effect of overabundant projection directions on 3D reconstruction algorithms. J. Struct. Biol. 133, 108–118 (2001)CrossRefGoogle Scholar
  59. 59.
    Udupa, J.K., Herman, G.T.: 3D Imaging in Medicine, 2nd edn. CRC Press, Boca Raton (1999)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Computer ScienceThe Graduate Center of the City University of New YorkNew YorkUSA

Personalised recommendations