Abstract
We give a very short account of ill-posed problems and the method of regularization. We then show how this method is being used in various problems from tomography, such as incomplete problems and the problem of attenuation correction in emission computed tomography.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Budinger, T.F. and Gullberg, G.T. (1979). Emission computed tomography. In: Image reconstruction from projections, G.T. Herman (ed.), Springer-Verlag, Berlin.
Buonocore, M.H., Brody, W.R., and Macowski, A. (1981). A natural pixel decomposition for two-dimensional image reconstructions, IEEE Trans. on Biom. Eng. BME-28, pp. 69–77.
Censor, Y., Gustafson, D.E., Lent, A., and Tuy, H. (1979). A new approach to the emission computerized tomography problem: simultaneous calculations of attenuation and activity coefficients, IEEE NS, Special Issue, April 1979.
Davison, M.E. (1983). The ill-conditioned nature of the limited angle tomography problem, SIAM J. Appl. Math. 43, pp. 428–448.
Gardner, R.J. and Mc Mullen, P. (1980). On Hammer’s X-ray Problem, J. London Math. Soc. 21, pp. 171–175.
Groetsch, C.W. (1983). The theory of Tikhonov regularization for Fredholm equations of the first Kind, Pitman, Boston.
Heike, U. (1984). Die Inversion der gedämpften Radon-Transformation, ein Rekonstruktionsverfahren der Emissionstomographie, Thesis, Universität Münster.
Helgason, S. (1980). The Radon transform, Birkhäuser, Boston.
Herman, G.T. (1980). Image reconstruction from projections, Academic Press, New York.
Lavrent’ev, M.M., Romanov, V.G., and Sisatskij, S.P. (1983). Problemi nonben posti in Fisica matematica ed Analisi,Conzilio Nazionale delle Ricerche, Publicazioni dell’Instituo di Analisi Globale e Applicazioni, Serie ‘Problemi non ben posti ed inversi’, n. 12, Florence.
Louis, A.K. (1980). Picture reconstruction from projections in limited range, Math. Meth. in the Appt. Sci. 2, pp. 209–220.
Natterer, F. (1980). Efficient implementation of ‘optimal’ algorithms in computerized tomography, Math. Meth. in the AppZ. Sci. 2, pp. 545–555.
Natterer, F. (1983). Computerized tomography with unknown sources, SIAM J.AppZ. Math. 43, pp. 1201–1212.
Natterer, F. (1986). The mathematics of computerized tomography, WileyTeubner, Stuttgart.
Peres, A. (1979). Tomographic reconstruction from limited angular data, J. Comput. Assist. Tomogr. 3, pp. 800–803.
Rangayan, R.M., Dhawan, A.P., and Gordon, R. (1985). Algorithms for limited-view tomography: an annotated bibliography and chalange, Appl. Optics 24, pp. 4000–4012.
Slepian, D. (1978). Prolate spheroidal wave functions, Fourier analysis, and uncertainty V: The discrete case, Bell. Syst. Techn. J. 57, pp. 1371–1430.
Tikhonov, A.N. and Arsenin, V.Y. (1977). Solution of ill-posed problems, Wiley, New York.
Tretiak, O. and Metz, C. (1980). The exponential Radon transform, SIAM J. Appl. Math. 39, pp. 341–354.
Volcic, A. (1983). Well-posedness of the Gardner-Mc Mullen reconstruction problem, Proceedings of the conference on measure theory, Springer-Verlag, Berlin, Lecture Notes in Mathematics 1089, pp. 199–210.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1988 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Natterer, F. (1988). Regularization Techniques in Medical Imaging. In: Viergever, M.A., Todd-Pokropek, A. (eds) Mathematics and Computer Science in Medical Imaging. NATO ASI Series, vol 39. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-83306-9_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-83306-9_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-83308-3
Online ISBN: 978-3-642-83306-9
eBook Packages: Springer Book Archive