Medical Imaging: State-of-the-Art and Future Development

  • A. K. Louis
Conference paper
Part of the Inverse Problems and Theoretical Imaging book series (IPTI)


The aim of medical imaging is to provide in an non - invasive way morphological information about a human patient. The information is obtained by performing an ” experiment ” where the interaction of a source of radiation anf the tissue under consideration is measured. From the measured data the desired information has to be computed, hence we face an inverse problem. It is always ill - posed in the sense that small errors in the data can be amplified to large changes in the reconstruction. For developing efficient and stable software we have to study the mathematical model; i. e., the description of the experiment based on physical and engeneering knowledge. In optimal situations it is possible to derive” inversion formulas” which relate in a constructive way the data to the searched - for information. Reconstruction algorithms can be found by discretisizing these formulas. But of course we have to perform a stability analysis in order to design the software such that the influence of the data noise is reduced as much as possible. If such inversion formulas are unknown or cannot be discretisized in an accurate way direct discretization and iterative methods are used for the computation.


Single Photon Emission Computerize Tomography Inverse Problem Inversion Formula Boundary Measurement Direct Discretization 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Dolveck - Guilpart, B.: Deux problèmes de représentation et d’identification d’un milieu a partir de sondages, Thesis, Montpellier, 1989Google Scholar
  2. [2]
    Greenleef, J.F.: Computerized tomography with ultrasound. Proc. IEEE 71, 356–372, 1983CrossRefGoogle Scholar
  3. [3]
    Grünbaum, F.A.: Reconstruction with arbitrary directions: dimensions two and three. In: Herman, G.T., Natterer, F.(eds.): Mathematical aspects of computerized tomography, Springer LNMI 8, 112–126, 1981Google Scholar
  4. [4]
    Herman, G.T.: Image reconstruction from projections. The fundamentals of computerized tomography. Academic Press, New York, 1980MATHGoogle Scholar
  5. [5]
    Hinshaw, W.S., Lent, A.H.: An introduction to NMR imaging: from Bloch equation to imaging equation, Proc. IEEE 71, 338–350, 1983CrossRefGoogle Scholar
  6. [6]
    Knoll, G.F.: Single-photon emission computed tomography. Proc. IEEE 71, 320–329, 1983CrossRefGoogle Scholar
  7. [7]
    Kremer, J., Louis, A.K.: About the mathematical foundation of hyperthermia treatment. Math. Meth. Appl. Sci. in press, 1990Google Scholar
  8. [8]
    Langenberg, K.J.: Introduction to the special issue of inverse problems. Wave Motion 11, 99–112, 1989CrossRefMATHGoogle Scholar
  9. [9]
    Lewitt, R.M.: Reconstruction algorithms: transform methods. Proc. IEEE 71, 390–408, 1983CrossRefGoogle Scholar
  10. [10]
    Louis, A.K.: Picture reconstruction from projections in restricted range. Math. Meth. Appl. Sci. 2, 209–220, 1980CrossRefMATHMathSciNetGoogle Scholar
  11. [11]
    Louis, A.K.: Incomplete data problems in x-ray computerized tomography, I: Singular value decomposition of the limited angle transform. Numer. Math. 48, 251–262, 1986CrossRefMATHMathSciNetGoogle Scholar
  12. [12]
    Louis, A.K.: Inverse und Schlecht Gestellte Probleme, Teubner, Stuttgart, 1989MATHGoogle Scholar
  13. [13]
    Louis, A.K., Maaß,P.: A mollifier method for linear operator equations of the first kind. Inverse Problems, to appearGoogle Scholar
  14. [14]
    Louis, A.K., Natterer, F.: Mathematical problems of computerized tomography. Proc. IEEE, 71, 379–389, 1983CrossRefGoogle Scholar
  15. [15]
    Louis, A.K., Rieder, A.: Incomplete data problems in x-ray computerized tomography, II: Truncated projections and region-of-interest tomography. Numer. Math. 56, 371–383, 1989CrossRefMATHMathSciNetGoogle Scholar
  16. [16]
    Louis, A.K., Schwierz, G.: Rekonstruktionsverfahren in der medizinischen Bildgebung. ZAMM 70 1990Google Scholar
  17. [17]
    Macovski, A.: Physical problems of computerized tomography. Proc. IEEE 71, 373–378, 1983CrossRefGoogle Scholar
  18. [18]
    Nachman, A.I.: Reconstructions from boundary measurements. Annals of Mathematics 128, 531–576, 1988CrossRefMATHMathSciNetGoogle Scholar
  19. [19]
    Natterer, F.: Computerized tomography with unknown sources. SIAMJ. Appl. Math. 43, 1201–1212, 1983MATHMathSciNetGoogle Scholar
  20. [20]
    Natterer, F.: The mathematics of computerized tomography. Teubner - Wiley, Stuttgart - New York, 1986MATHGoogle Scholar
  21. [21]
    Ramm, A.G.: Recovery of the potential from fixed-energy scattering data. Inverse Problems 4, 877–886, 1988CrossRefMATHADSMathSciNetGoogle Scholar
  22. [22]
    Sabatier, P.C.: Tomography and inverse problems. Adam Hilger, Bristol, 1987MATHGoogle Scholar
  23. [23]
    Santosa, F., Vogelius, M.: A backprojection algorithm for electrical impedance imaging. SIAM J. Appl. Math.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • A. K. Louis
    • 1
  1. 1.Fachbereich MathematikTechnische Universität BerlinGermany

Personalised recommendations