Abstract
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.
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References
Dolveck - Guilpart, B.: Deux problèmes de représentation et d’identification d’un milieu a partir de sondages, Thesis, Montpellier, 1989
Greenleef, J.F.: Computerized tomography with ultrasound. Proc. IEEE 71, 356–372, 1983
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, 1981
Herman, G.T.: Image reconstruction from projections. The fundamentals of computerized tomography. Academic Press, New York, 1980
Hinshaw, W.S., Lent, A.H.: An introduction to NMR imaging: from Bloch equation to imaging equation, Proc. IEEE 71, 338–350, 1983
Knoll, G.F.: Single-photon emission computed tomography. Proc. IEEE 71, 320–329, 1983
Kremer, J., Louis, A.K.: About the mathematical foundation of hyperthermia treatment. Math. Meth. Appl. Sci. in press, 1990
Langenberg, K.J.: Introduction to the special issue of inverse problems. Wave Motion 11, 99–112, 1989
Lewitt, R.M.: Reconstruction algorithms: transform methods. Proc. IEEE 71, 390–408, 1983
Louis, A.K.: Picture reconstruction from projections in restricted range. Math. Meth. Appl. Sci. 2, 209–220, 1980
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, 1986
Louis, A.K.: Inverse und Schlecht Gestellte Probleme, Teubner, Stuttgart, 1989
Louis, A.K., Maaß,P.: A mollifier method for linear operator equations of the first kind. Inverse Problems, to appear
Louis, A.K., Natterer, F.: Mathematical problems of computerized tomography. Proc. IEEE, 71, 379–389, 1983
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, 1989
Louis, A.K., Schwierz, G.: Rekonstruktionsverfahren in der medizinischen Bildgebung. ZAMM 70 1990
Macovski, A.: Physical problems of computerized tomography. Proc. IEEE 71, 373–378, 1983
Nachman, A.I.: Reconstructions from boundary measurements. Annals of Mathematics 128, 531–576, 1988
Natterer, F.: Computerized tomography with unknown sources. SIAMJ. Appl. Math. 43, 1201–1212, 1983
Natterer, F.: The mathematics of computerized tomography. Teubner - Wiley, Stuttgart - New York, 1986
Ramm, A.G.: Recovery of the potential from fixed-energy scattering data. Inverse Problems 4, 877–886, 1988
Sabatier, P.C.: Tomography and inverse problems. Adam Hilger, Bristol, 1987
Santosa, F., Vogelius, M.: A backprojection algorithm for electrical impedance imaging. SIAM J. Appl. Math.
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© 1990 Springer-Verlag Berlin Heidelberg
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Louis, A.K. (1990). Medical Imaging: State-of-the-Art and Future Development. In: Sabatier, P.C. (eds) Inverse Methods in Action. Inverse Problems and Theoretical Imaging. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-75298-8_3
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DOI: https://doi.org/10.1007/978-3-642-75298-8_3
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