The Identification Problem in Emission Computed Tomography

  • F. Natterer
Part of the Lecture Notes in Medical Informatics book series (LNMED, volume 8)

Summary

In (single photon) emission computed tomography one has to compute the activity distribution f(x) and the distribution of the attenuation coefficient μ (x) from the integral equation
$$\mathop \smallint \limits_{L(s,\omega )} f(x){e^{ - (M\mu )(x,\omega )}}dx = g(s,w)$$
where ω is a unit vector, L(s, ω) is the straight line perpendicular to a ω with (signed) distance s from the origin, and
$$(M\mu )(x,\omega ) = \mathop \smallint \limits_o^\infty \mu (x + t{\omega ^ - })dt$$
is the fan beam transform. We show that for finitely many sources, Mμ (x,ω) can be determined up to an additive constant for all sources x by the consistency conditions in the range of the attenuated Radon transform.

Keywords

Attenuation Radon 

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References

  1. [1]
    Budinger, T. F. - Gullberg, G. T. - Huesman, R. H.: Emission Computed Tomography, in: G. T. Herman (ed.): Image Reconstruction from Projections. Topics in Applied Physics, vol. 32, Springer 1979Google Scholar
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    Tretiak, O. - Metz, C.: The Exponential Radon Transform, Manuscript, Drexel UniversityGoogle Scholar
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    Censor, Y. - Gustafson, D. E. - Lent, A. - Tuy, H.: A New Approach to the Emission Computerized Tomography Problem: Simultaneous Calculation of Attenuation and Activity Coefficients, IEEE NS, Special Issue, April 1979Google Scholar
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    Natterer, F.: The Ill-Posedness of Radon’s Integral Equation, Symposion on Ill-Posed Problems: Theory and Practice,Delaware, USA, October 2–6, 1979Google Scholar
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    Hamaker, C. - Smith, K. T. - Solmon, D. C. - Wagner, S. L.: The Divergent Beam X-Ray Transform, to appearGoogle Scholar
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    Ludwig, D.: The Radon Transform in Euclidean Snace, Comm. Pure Appl. Math. 19, 48–81 (1966)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1981

Authors and Affiliations

  • F. Natterer
    • 1
  1. 1.Fachbereich 10Universität des SaarlandesSaarbrückenDeutschland

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