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Numerical methods in tomography

Published online by Cambridge University Press:  07 November 2008

Frank Natterer
Frank Natterer
Affiliation:
Institut für Numerische und Instrumentelle Mathematik, Universität Münster, Einsteinstrasse 62, D-48149 Münster, Germany E-mail: nattere@math.uni-muenster.de
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Abstract

In this article we review the image reconstruction algorithms used in tomography. We restrict ourselves to the standard problems in the reconstruction of function from line or plane integrals as they occur in X-ray tomography, nuclear medicine, magnetic resonance imaging, and electron microscopy. Nonstandard situations, such as incomplete data, unknown orientations, local tomography, and discrete tomography are not dealt with. Nor do we treat nonlinear tomographic techniques such as impedance, ultrasound, and near-infrared imaging.

Type
Research Article
Information
Acta Numerica , Volume 8 , January 1999 , pp. 107 - 141
Copyright
Copyright © Cambridge University Press 1999

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References

REFERENCES

Abramowitz, M. and Stegun, I. A., eds (1970), Handbook of Mathematical Functions, Dover.Google Scholar
Censor, Y. (1981), ‘Row-action methods for huge and sparse systems and their applications’, SIAM Review 23, 444–466.CrossRefGoogle Scholar
Censor, Y., Eggermont, P. B. and Gordon, D. (1983), ‘Strong underrelaxation in Kaczmarz's method for inconsistent systems’, Numer. Math. 41, 83–92.Google Scholar
Censor, Y. and Zenios, S. A. (1997), Parallel Optimization, Oxford University Press.Google Scholar
Chang, L. T. and Herman, G. T. (1980), ‘A scientific study of filter selection for a fan-beam convolution algorithm’, SIAM J. Appl. Math. 39, 83–105.CrossRefGoogle Scholar
Chapman, C. H. and Cary, P. W. (1986), ‘The circular harmonic Radon transform’, Inverse Problems 2, 23–49.Google Scholar
Colsher, J. G. (1980), ‘Fully three-dimensional emission tomography’, Phys. Med. Biol. 25, 103–115.CrossRefGoogle ScholarPubMed
Cormack, A. M. (1963), ‘Representation of a function by its line integrals, with some radiological applications I’, J. Appl. Phys. 34, 2722–2727.CrossRefGoogle Scholar
Cormack, A. M. (1964), ‘Representation of a function by its line integrals, with some radiological applications II’, J. Appl. Phys. 35, 195–207.CrossRefGoogle Scholar
Deans, S. R. (1983), The Radon Transform and some of its Applications, Wiley.Google Scholar
Defrise, M., Townsend, D. W. and Clack, R. (1989), ‘Three-dimensional image reconstruction from complete projections’, Phys. Med. Biol. 34, 573–587.CrossRefGoogle ScholarPubMed
Defrise, M. and Clack, R. (1995), ‘A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection’, IEEE Trans. Med. Imag. 13, 186–195.Google Scholar
Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977), ‘Maximum likelihood from incomplete data via the EM algorithm’, J. R. Statist. Soc. B 39, 1–38.Google Scholar
Edholm, P. and Herman, G. T. (1987), ‘Linograms in image reconstruction from projections’, IEEE Trans. Med. Imag. 6, 301–307.CrossRefGoogle ScholarPubMed
Faridani, A., Finch, D. V., Ritman, E. L. and Smith, K. T. (1997), ‘Local tomography II’, SIAM J. Appl. Math. 57, 1095–1127.Google Scholar
Frank, J., ed. (1992), Electron Tomography, Plenum Press.Google Scholar
Gelfand, I. M. and Goncharov, A. B. (1987), ‘Recovery of a compactly supported function starting from its integrals over lines intersecting a given set of points in space’, Doklady 290 (1986); English Translation in Soviet Math. Dokl. 34, 373–376.Google Scholar
Gelfand, I. M. and Goncharov, A. B. (1990), ‘Spatial rotational alignment of identical particles given their projections: Theory and practice’, Translation of Mathematical Monographs 81, 97–122.Google Scholar
Gordon, R., Bender, R. and Herman, G. T. (1970), ‘Algebra reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography’, J. Theor. Biol. 29, 471–481.Google Scholar
Grangeat, P. (1991), ‘Mathematical framework of cone-beam reconstruction via the first derivative of the Radon transform’, in Mathematical Methods in Tomography, Vol. 1497 of Lecture Notes in Mathematics (Herman, G. T., Louis, A. K. and Natterer, F., eds), Springer, pp. 66–97.Google Scholar
Green, P. J. (1990), ‘Bayesian reconstruction from emission tomography data using a modified EM algorithm’, IEEE Trans. Med. Imag. 9, 84–93.CrossRefGoogle ScholarPubMed
Hamaker, C. and Solmon, D. C. (1978), ‘The angles between the null spaces of X-rays’, J. Math. Anal. Appl. 62, 1–23.CrossRefGoogle Scholar
Hansen, E. W. (1981), ‘Circular harmonic image reconstruction’, Applied Optics 20, 2266–2274.Google Scholar
Hawkins, W. G. and Barrett, H. H. (1986), ‘A numerically stable circular harmonic reconstruction algorithm’, SIAM J. Numer. Anal. 23, 873–890.Google Scholar
Herman, G. T. (1980), Image Reconstruction from Projection: The Fundamentals of Computerized Tomography, Academic Press.Google Scholar
Herman, G. T. and Meyer, L. (1993), ‘Algebraic reconstruction techniques can be made computationally efficient’, IEEE Trans. Med. Imag. 12, 600–609.Google Scholar
Hudson, H. M., Hutton, B. F. and Larkin, R. (1992), ‘Accelerated EM reconstruction using ordered subsets’, J. Nucl. Med. 33, 960–968.Google Scholar
Jerry, A. J. (1977), ‘The Shannon sampling theorem - its various extensions and applications: a tutorial review’, Proc. IEEE 65, 1565–1596.Google Scholar
Kaczmarz, S. (1937), ‘Angenäherte Auflösung von Systemen linearer Gleichungen’, Bulletin de l'Académie Polonaise des Sciences et des Lettres A35, 355–357.Google Scholar
Kak, A. C. and Slaney, M. (1987), Principles of Computerized Tomography Imaging, IEEE Press, New York.Google Scholar
Kaveh, M. and Soumekh, M. (1987), ‘Computer assisted diffraction tomography’, in Image Recovery: Theory and Application (Stark, H., ed.), Academic Press, pp. 369–413.Google Scholar
Kruse, H. (1989), ‘Resolution of reconstruction methods in computerized tomography’, SIAM J. Sci. Statist. Comput. 10, 447–474.Google Scholar
Levitan, E. and Herman, G. T. (1987), ‘A maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography’, IEEE Trans. Med. Imag. 6, 185–192.Google Scholar
Lewitt, R. M. (1992), ‘Alternatives to voxels for image representations in iterative reconstruction algorithms’, Phys. Med. Biol. 37, 705–716.CrossRefGoogle ScholarPubMed
Louis, A. K. (1980), ‘Picture reconstruction from projections in restricted range’, Math. Meth. Appl. Sci. 2, 109–220.CrossRefGoogle Scholar
Marabini, R., Herman, G. T. and Carazo, J. M. (1998), ‘Fully three-dimensional reconstruction in electron microscopy’, in Computational Radiology and Imaging: Therapy and Diagnostics (Borgers, C. and Natterer, F., eds), Vol. 110 of IMA Volumes in Mathematics and its Applications, Springer.Google Scholar
Marr, R. B., Chen, C. N. and Lauterbur, P. C. (1981), ‘On two approaches to 3D reconstruction in NMR zeugmatography’, in Mathematical Aspects of Computerized Tomography (Herman, G. T. and Natterer, F., eds), Proceedings, Oberwolfach 1980, Springer, pp. 225–240.Google Scholar
Natterer, F. (1986), The Mathematics of Computerized Tomography, Wiley and Teubner.Google Scholar
Natterer, F. (1993), ‘Sampling in fan beam tomography’, SIAM J. Appl. Math. 53, 358–380.Google Scholar
Natterer, F. and Faridani, A. (1990), ‘Basic algorithms in tomography’, in Signal Processing Part II: Control Theory and Applications (Grünbaum, F. A. et al. , eds), Springer, pp. 321–334.Google Scholar
Nilsson, S. (1997), ‘Application of fast backprojection techniques for some inverse problems of integral geometry’, Linköping Studies in Science and Technology, Dissertation No. 499, Department of Mathematics, Linköping University, Linköping, Sweden.Google Scholar
Nussbaumer, H. J. (1982), Fast Fourier Transform and Convolution Algorithm, Springer.Google Scholar
Orlov, S. S. (1976), ‘Theory of three dimensional reconstruction, II: The recovery operator’, Sov. Phys. Crystallogr. 20, 429–433.Google Scholar
O'Sullivan, J. D. (1985), ‘A fast sinc function gridding algorithm for Fourier inversion in computer tomography’, IEEE Trans. Med. Imag. 4, 200–207.CrossRefGoogle ScholarPubMed
De Pierro, A. R. (1990), ‘Multiplicative interative methods in computed tomography’, in Mathematical Methods in Tomography (Herman, G. T., Louis, A. K. and Natterer, F., eds), Springer, pp. 167–186.Google Scholar
Radon, J. (1917), ‘Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten’, Berichte Sächsische Akademie der Wissenschaften, Math.-Phys. Kl., 69, 262–267, Leipzig.Google Scholar
Ramm, A. and Katsevich, A. (1996), The Radon Transform and Local Tomography, CRC Press.Google Scholar
Schomberg, H. and Timmer, J. (1995), ‘The gridding method for image reconstruction bx Fourier transformation’, IEEE Trans. Med. Imag. 14, 596–607.Google Scholar
Shepp, L. A. and Logan, B. F. (1974), ‘The Fourier reconstruction of a head section’, IEEE Trans. Trans. Nucl. Sci. NS-21, 21–43.CrossRefGoogle Scholar
Shepp, L. A. and Vardi, Y. (1982), ‘Maximum likelihood reconstruction for emission tomography’, IEEE Trans. Med. Imag. 1, 113–122.CrossRefGoogle ScholarPubMed
Setzepfandt, B. (1992), ‘ESNM: Ein rauschunterdrückendes EM-Verfahren für die Emissionstomographie’, Dissertation, Fachbereich Mathematik, Universität Münster.Google Scholar
Sielschott, H. and Derichs, W. (1995), ‘Use of collocation methods under inclusion of a priori information in acoustic pyrometry’, Proc. European Concerted Action on Process Tomography, Bergen, Norway, pp. 110–117.Google Scholar
Silverman, B. W., Jones, M. C., Nychka, D. W. and Wilson, J. D. (1990), ‘A smoothed EM approach to indirect estimation problems, with particular reference to stereology and emission tomography’, J. R. Statist. Soc. B 52, 271–324.Google Scholar
Tuy, H. K. (1983), ‘An inversion formula for cone-beam reconstruction’, SIAM J. Appl. Math. 43, 546–552.Google Scholar
Welch, A., Clack, R., Natterer, F. and Gullberg, G. T. (1997), ‘Towards accurate attenuation correction in SPECT’, IEEE Trans. Med. Imag. 16, 532–541.Google Scholar
Wuschke, K. (1990), ‘Die Rekonstruktion von Orientierungen aus Projektionen’, Diplomarbeit, Institut für Numerische und Instrumentelle Mathematik, Universität Münster.Google Scholar