Abstract
Many inverse problems in medicine, and in particular in tomography, are ill-posed, that is, the result of the inversion is unstable with respect to small perturbations of the measured data. This paper introduces the concept of ill-posed problem, and describes several classical regularization techniques, which allow one to recover stable solutions to ill-posed inverse problems. Applications to the problem of image reconstruction from projections are presented in the last section of the paper.
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
Bakushinskii, A.B. (1967). A general method of constructing regularizing algorithms for a linear ill-posed equation in Hilbert space, U.S.S.R Comp. Maths. Math. Phys. 7, pp. 279–287.
Barrett H.H. and Swindell W. (1981). Radiological Imaging. Academic Press.
Bertero M., De Mol C., Viano G. (1980). The Stability of Inverse Problems. In Inverse Scattering in Optics (H.P. Baltes ed.), Topics in Current Physics 20, Springer-Verlag, pp. 161-214.
Budinger T.F., Gullberg G.T. and Huesman R.H. (1979). Emission Computed Tomography, in Image Reconstruction from Projections (Herman G.T. ed). Topics in Applied Physics, Springer.
ai]Davison, M.E. (1983). The ill-conditioned nature of the limited angle tomography problem. SIAM J. Appl. Math. 43, pp. 428–448.
Defrise M., De Mol C. (1984). Resolution limits in full-and limited angle tomography. Proceedings of the 8th conference on Information Processing in Medical Imaging, F. Deconinck ed., Martinus Nijhoff, pp.94-105.
Defrise M., De Mol C. (1987). A note on stopping rules for iterative regularization methods and filtered SVD, in “Inverse problems, an interdisciplinary study” (P.C. Sabatier ed.), Advances in Electronics and Electron Physics, supplement 19, Academic Press, pp. 261-268.
Hadamard J. (1923). Lectures on the Cauchy problem in linear partial differential equations. Yale University Press, New Haven.
Lim C.B., Han K.S., Hawman E.G. and Jaszczak R.J. (1982). Image noise, resolution, and lesion detectability in SPECT. IEEE Trans. Nucl. Sc. NS-29, pp.500-505.
Louis A.K. (1980). Picture reconstruction from projections in a restricted range. Math. Meth. in Appl. Sc. 2, pp. 209–220.
Miller K. (1970). Least squares methods for ill-posed problems with a prescribed bound. SIAM J. Math. Anal. 1, pp. 52–74.
Morozov V.A. (1968). The error principle in the solution of operational equations by the regularization method. USSR Comp. Maths. Math. Phys. 8, pp. 63.
Natterer F. (1986). The mathematics of computerized tomography. Wiley & Sons.
Slepian D. (1978). Prolate spheroidal wave functions, Fourier analysis and uncertainty. V. The discrete case. Bell Sys. Tech. J. 57, pp.1371–1430.
Tikhonov A. and Arsenine V. (1977). Solution of ill-posed problems. Winston &Sons.
Vainikko G.M. (1983). The critical level of discrepancy in regularization methods. USSR Comp. Maths. Math. Phys. 23, pp.1–9.
Veklerov E. and Llacer J. (1987). Stopping rule for the MLE algorithm based on a statistical hypothesis testing. IEEE Trans. Med. Imag. MI-6, pp.313–319.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Defrise, M. (1992). Regularization Techniques in Medical Imaging. In: Todd-Pokropek, A.E., Viergever, M.A. (eds) Medical Images: Formation, Handling and Evaluation. NATO ASI Series, vol 98. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-77888-9_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-77888-9_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-77890-2
Online ISBN: 978-3-642-77888-9
eBook Packages: Springer Book Archive