Medical Imaging: State-of-the-Art and Future Development
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.
KeywordsSingle Photon Emission Computerize Tomography Inverse Problem Inversion Formula Boundary Measurement Direct Discretization
Unable to display preview. Download preview PDF.
- Dolveck - Guilpart, B.: Deux problèmes de représentation et d’identification d’un milieu a partir de sondages, Thesis, Montpellier, 1989Google Scholar
- 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
- Kremer, J., Louis, A.K.: About the mathematical foundation of hyperthermia treatment. Math. Meth. Appl. Sci. in press, 1990Google Scholar
- Louis, A.K., Maaß,P.: A mollifier method for linear operator equations of the first kind. Inverse Problems, to appearGoogle Scholar
- Louis, A.K., Schwierz, G.: Rekonstruktionsverfahren in der medizinischen Bildgebung. ZAMM 70 1990Google Scholar
- Santosa, F., Vogelius, M.: A backprojection algorithm for electrical impedance imaging. SIAM J. Appl. Math.Google Scholar