Algebraic Reconstruction of Source and Attenuation in SPECT Using First Scattering Measurements

  • Evelyn Cueva
  • Axel Osses
  • Juan Carlos Quintana
  • Cristián Tejos
  • Matías Courdurier
  • Pablo Irarrazaval
Part of the Trends in Mathematics book series (TM)


Here we present an Algebraic Reconstruction Technique (ART) for solving the identification problem in Single Photon Emission Computed Tomography (SPECT). Traditional reconstruction for SPECT is done by finding the radiation source, nevertheless the attenuation of the surrounding tissue affects the data. In this context, ballistic and first scattering information are used to recover source and attenuation simultaneously. Both measurements are related with the Attenuated Radon Transform and a Klein-Nishina angular type dependency is considered for the scattering. The proposed ART algorithm allow us to obtain good reconstructions of both objects in a few number of iterations.



E.C. was supported by CONICYT-PCHA/Doctorado Nacional/2016-21161721, A.O. was partially supported by CONICYT-Fondecyt grant 1151512. C.T. was partially supported by CONICYT - PIA - Anillo ACT1416. M.C. was partially supported by CONICYT - PIA - Anillo ACT1416 and Fondecyt grant number 1141189. P.I. was partially supported by CONICYT - PIA - Anillo ACT1416.


  1. 1.
    A.V. Bronnikov, Numerical solution of the identification problem for the attenuated Radon transform. Inverse Prob. 15(5), 1315 (1999)Google Scholar
  2. 2.
    A.V. Bronnikov, Reconstruction of attenuation map using discrete consistency conditions. IEEE Trans. Med. Imaging 19(5), 451–462 (2000)CrossRefGoogle Scholar
  3. 3.
    Y. Censor, D.E. Gustafson, A. Lent, H. Tuy, A new approach to the emission computerized tomography problem: simultaneous calculation of attenuation and activity coefficients. IEEE Trans. Nucl. Sci. 26(2), 2775–2779 (1979)CrossRefGoogle Scholar
  4. 4.
    M. Courdurier, F. Monard, A. Osses, F. Romero, Simultaneous source and attenuation reconstruction in SPECT using ballistic and single scattering data. Inverse Prob. 31(9), 095002 (2015)Google Scholar
  5. 5.
    V. Dicken, A new approach towards simultaneous activity and attenuation reconstruction in emission tomography. Inverse Prob. 15(4), 931 (1999)Google Scholar
  6. 6.
    D. Gourion, D. Noll, The inverse problem of emission tomography. Inverse Prob. 18(5), 1435 (2002)Google Scholar
  7. 7.
    S. Luo, J. Qian, P. Stefanov, Adjoint state method for the identification problem in SPECT: recovery of both the source and the attenuation in the attenuated X-ray transform. SIAM J. Imag. Sci. 7(2), 696–715 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    S.H. Manglos, Determination of the attenuation map from SPECT projection data alone. J. Nucl. Med. 35, 193 (1993)Google Scholar
  9. 9.
    R. Ramlau, R. Clackdoyle, Accurate attenuation correction in SPECT imaging using optimization of bilinear functions and assuming an unknown spatially-varying attenuation distribution. In Nuclear Science Symposium, 1998. Conference Record. 1998, vol. 3 (IEEE, New York, 1998), pp. 1684–1688Google Scholar
  10. 10.
    A. Welch, R. Clack, F. Natterer, G.T. Gullberg, Toward accurate attenuation correction in SPECT without transmission measurements. IEEE Trans. Med. Imaging 16(5), 532–541 (1997)CrossRefGoogle Scholar
  11. 11.
    H. Zaidi, B. Hasegawa, Determination of the attenuation map in emission tomography. J. Nucl. Med. 44(2), 291–315 (2003)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Evelyn Cueva
    • 1
  • Axel Osses
    • 1
  • Juan Carlos Quintana
    • 2
  • Cristián Tejos
    • 3
  • Matías Courdurier
    • 4
  • Pablo Irarrazaval
    • 5
  1. 1.Departamento de Ingeniería Matemática and Center for Mathematical ModelingUniversidad de ChileSantiagoChile
  2. 2.Departamento de RadiologíaPontificia Universidad Católica de ChileSantiagoChile
  3. 3.Departamento de Ingeniería EléctricaPontificia Universidad Católica de ChileSantiagoChile
  4. 4.Departamento de MatemáticasPontificia Universidad Católica de ChileSantiagoChile
  5. 5.Departamento de Ingeniería Eléctrica and Instituto de Ingeniería Biológica y MédicaPontificia Universidad Católica de ChileSantiagoChile

Personalised recommendations