Emission Discrete Tomography

  • E. Barcucci
  • A. Frosini
  • A. Kuba
  • A. Nagy
  • S. Rinaldi
  • M. Šámal
  • S. Zopf
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


Three problems of emission discrete tomography (EDT) are presented. The first problem is the reconstruction of measurable plane sets from two absorbed projections. It is shown that Lorentz theorems can be generalized to this case. The second is the reconstruction of binary matrices from their absorbed row and columns sums if the absorption coefficient is μ0 = log((1+v/5)/2). It is proved that the reconstruction in this case can be done in polynomial time. Finally, a possible application of EDT in single photon emission computed tomography (SPECT) is presented: Dynamic structures are reconstructed after factor analysis.


Single Photon Emission Compute Tomography Renal Pelvis Binary Matrix Generalize Projection Vertical Projection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Backfrieder, W., Samal, M., Bergmann, H.: Toward estimation of compartment volumes and radionuclide concentrations in dynamic SPECT using factor analysis and limited number of projections. Physica Medica, 15, 160 (1999).Google Scholar
  2. 2.
    Balogh, E., Kuba, A., Del Lungo, A., Nivat, M.: Reconstruction of binary matrices from absorbed projections. In: Braquelaire, A., Lachaud, J.-O., Vialard, A. (eds.), Discrete Geometry in Computer Imagery, Springer, Berlin, Germany, pp. 392–403 (2002).CrossRefGoogle Scholar
  3. 3.
    Barcucci, E., Frosini, A., Rinaldi, S.: Reconstruction of discrete sets from two absorbed projections: An algorithm. Electr. Notes Discr. Math., 12 (2003).Google Scholar
  4. 4.
    Frosini, A., Barcucci, E., Rinaldi, S.: An algorithm for the reconstruction of discrete sets from two projections in present of absorption. Discr. Appl. Math., 151, 21–35 (2005).MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Frosini, A., Rinaldi, S., Barcucci, E., Kuba, A.: An efficient algorithm for reconstructing binary matrices from horizontal and vertical absorbed projections. Electr. Notes Discr. Math., 20, 347–363 (2005).CrossRefMathSciNetGoogle Scholar
  6. 6.
    Gardner, R.J.: Geometric Tomography. Cambridge University Press, Cambridge, UK (1995).MATHGoogle Scholar
  7. 7.
    Hajdu, L., Tijdeman, R.: Algebraic aspects of emission tomography with absorption. J. Reine Angew. Math., 534, 119–128 (2001).MATHMathSciNetGoogle Scholar
  8. 8.
    Herman, G.T., Kuba, A. (eds.): Discrete Tomography: Foundations, Algorithms, and Applications. Birkhäuser, Boston, MA (1999).MATHGoogle Scholar
  9. 9.
    Kaneko, A., Huang, L.: Reconstruction of plane figures from two projections. In: Herman, G.T., Kuba, A. (eds.), Discrete Tomography. Foundations, Algorithms, and Applications. Birkäuser, Boston, MA, pp. 115–135 (1999).Google Scholar
  10. 10.
    Kellerer, H.: Masstheoretische Marginalprobleme. Math. Ann., 153, 168–198 (1964).MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kuba, A.: Reconstruction of two-directionally connected binary patterns from their two orthogonal projections. Comp. Vision Graph. Image Proc., 27, 249–265 (1984).CrossRefGoogle Scholar
  12. 12.
    Kuba, A.: Reconstruction of measurable sets from two absorbed projections. Techn. Rep., University of Szeged, Szeged, Hungary (2004).Google Scholar
  13. 13.
    Kuba, A., Nagy, A., Balogh, E.: Reconstruction of hv-convex binary matrices from their absorbed projections. Discr. Appl. Math., 139, 137–148 (2004).MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kuba, A., Nivat, M.: Discrete tomography with absorption. In: Borgefors, G., di Baja, S. (eds.), Discrete Geometry in Computer Imagery, Springer, Berlin, Germany, pp. 3–34 (2000).Google Scholar
  15. 15.
    Kuba, A., Nivat, M.: Reconstruction of discrete sets with absorption. Lin. Algebra Appl., 339, 171–194 (2001).MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kuba, A., Nivat, M.: A sufficient condition for non-uniqueness in binary tomography with absorption. Discr. Appl. Math., 346, 335–357 (2005).MATHMathSciNetGoogle Scholar
  17. 17.
    Kuba, A., Volcic, A.: Characterization of measurable plane sets which are reconstructable from their two projections. Inverse Problems, 4, 513–527 (1988).MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kuba, A., Volcic, A.: The structure of the class of non-uniquely reconstructible sets. Acta Sci. Szeged, 58, 363–388 (1993).MATHMathSciNetGoogle Scholar
  19. 19.
    Lorentz, G.G.: A problem of plane measure. Amer. J. Math., 71, 417–426 (1949).MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Metropolis, N., Rosenbluth, A., Rosenbluth, A.T.M., Teller, E.: Equation of state calculation by fast computing machines. J. Chem. Phys., 21, 1087–1092 (1953).CrossRefGoogle Scholar
  21. 21.
    Nagy, A., Kuba, A., Samal, M.: Reconstruction of factor structures using discrete tomography method. Electr. Notes Comput. Sci., 20, 519–534 (2005).MathSciNetGoogle Scholar
  22. 22.
    Reeds, J.A., Shepp, L.A., Fishburn, P.C., Lagarias, J.C.: Sets uniquely determined by projections. I. Continuous case. SIAM J. Applied Math., 50, 288–306 (1990).MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Ryser, H.R.: Combinatorial properties of matrices of zeros and ones. Canad. J. Math., 9, 371–377 (1957).MATHMathSciNetGoogle Scholar
  24. 24.
    Samal, M., Karny, M., Surova, H., Marikova, E., Dienstbier, Z.: Rotation to simple structure in factor analysis of dynamic radionuclide studies. Phys. Med. Biol., 32, 371–382 (1987).CrossRefGoogle Scholar
  25. 25.
    Samal, M., Nimmon, C.C., Britton, K.E., Bergmann, H.: Relative renal uptake and transit time measurements using functional factor images and fuzzy regions of interest. Eur. J. Nucl. Med., 25, 48–54 (1998).Google Scholar
  26. 26.
    Slicer program: http://www.slicer.org.
  27. 27.
    Zopf, S., Kuba, A.: Reconstruction of measurable sets from two generalized projections. Electr. Notes Discr. Math., 20, 47–66 (2005).CrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston 2007

Authors and Affiliations

  • E. Barcucci
    • 1
  • A. Frosini
    • 2
  • A. Kuba
    • 3
  • A. Nagy
    • 3
  • S. Rinaldi
    • 2
  • M. Šámal
    • 4
  • S. Zopf
    • 3
  1. 1.Dipartimento di Sistemi e InformaticaUniversit’a degli Studi di FirenzeFirenzeItaly
  2. 2.Dip. di Scienze Matematiche e InformaticheUniv. degli Studi di SienaSienaItaly
  3. 3.Dept. of Image Processing and Computer GraphicsUniversity of SzegedSzegedHungary
  4. 4.Dept. of Nuclear MedicineFirst Faculty of Medicine, Charles Univ. PraguePraha 2Czech Republic

Personalised recommendations