A Recursive Algorithm for Diffuse Planar Tomography

  • Sarah K. Patch
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


Diffuse tomography generalizes the standard discrete tomography problem,permitting anisotropic scattering. The diffuse problem involves more unknowns, and also more data. Markov transition probabilities are recovered from measurements taken at all pairs of input/output ports on the boundary. A recursive algorithm is used to solve the problem on a general n × n lattice in the plane.


Inverse Problem Imaging Object Hide State Recursive Algorithm Range Condition 
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.


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  1. [1]
    J. Singer, F. A. C, P. Kohn, and J. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).CrossRefPubMedGoogle Scholar
  2. [2]
    G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography, (Academic Press, New York), 1980.Google Scholar
  3. [3]
    S. K. Patch, “Consistency conditions in diffuse tomography,” Inverse Problems 10, 199–212 (1994).CrossRefGoogle Scholar
  4. [4]
    S. K. Patch, “Iterative algorithm for discrete tomography,” International Journal of Imaging Systems and Technology 9, 132–134 (1998).CrossRefGoogle Scholar
  5. [5]
    S. K. Patch, “Recursive recovery of a family of Markov transition probabilities from boundary value data,” Journal of Mathematical Physics 36 3395–3412 (1995).CrossRefGoogle Scholar
  6. [6]
    F. A. Grünbaum and S. K. Patch, “How many parameters can one solve for in Diffuse Tomography,” In G. Papanicolaou, A. Friedman, and R. Gulliver I.M.A. Workshop on Inverse Problems in Waves and Scattering, (Springer-Verlag, New York), pp. 219–236, 1995.Google Scholar
  7. [7]
    F. A. Grünbaum and S. K. Patch, “The use of Graßmann identities for inversion of a general model in diffuse tomography,” In Proceedings of the Lapland Conference on Inverse Problems, (Saariselkä, Finland), 1992.Google Scholar
  8. [8]
    S. K. Patch, “Diffuse tomography modulo Graßmann and Laplace,” Journal of Mathematical Physics 37, 3283–3305 (1996).CrossRefGoogle Scholar
  9. [9]
    W. V. D. Hodge and D. Pedoe, Methods of Algebraic Geometry, (Cambridge University Press, Cambridge, UK) 1968.Google Scholar
  10. [10]
    P. Griffiths and J. Harris, Principles of Algebraic Geometry, (Wiley and Sons, New York) 1978.Google Scholar
  11. [11]
    S. K. Patch, “Sufficiency and simplicity of range conditions for diffuse tomgraphic systems,” Technical Report #97–07, Scientific Computing and Computational Mathematics, Stanford University, Stanford, CA (1997).Google Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Sarah K. Patch
    • 1
  1. 1.General Electric Company Corporate Research and DevelopmentIndustrial Electronics LaboratorySchenectadyUSA

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