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Exploiting the Ranges of Radon Transformsin Tomography

  • Frank Natterer
Part of the Progress in Scientific Computing book series (PSC, volume 2)

Abstract

The classical Radon transform of a function f with support in the unit disc Ω of IR2 is defined as
$${\rm{Rf(s,\omega )}}\;{\rm{ = }}\;\int\limits_{{\rm{x}}{\rm{\omega }} = {\rm{s}}} {{\rm{f(x)dx}}\;{\rm{ = }}} \;\int {{\rm{f(s\omega }}\;{\rm{ + }}\;{\rm{t}}{{\rm{\omega }}^{\rm{\perp }}}{\rm{)}}\;{\rm{dt}}}$$
Here, s ε IR1 and ω= (cos ϕ, s in ϕ), ω = (−sin ϕ, cos ϕ), o ≤ ϕ < 2π. Thus, R associates with each f the set of its line integrals.

Keywords

Chebyshev Polynomial Homogeneous Polynomial Source Distribution Line Integral Radon Transform 
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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References

  1. [1]
    Cormack, A.M.: Representation of a Function by its Line Integrals, with same Radiological Applications I, J. Appl. Physics 34, 2722–2727.Google Scholar
  2. [2]
    Davison, M.E.: The Ill-Conditioned Nature of the Limited Angle Tomography Problem. To appear in SIAM J. Appl. Math.Google Scholar
  3. [3]
    Helgason, S.: The Radon Transform. Birkhäuser 1980.Google Scholar
  4. [4]
    Herman, G.T. (ed.): Image Reconstruction from Projection: Implementation and Applications. Springer 1979.Google Scholar
  5. [5]
    Louis, A.K.: Picture Reconstruction from Projections in Limited Range, Math. Meth. in the Appl. Sci. 2, 209–220 (1980).MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Natterer, F.: The Ill-Posedness of Radon’s Integral Equations, in: Proceedings of the Conference on Ill-Posed Problems, Delaware 1979.Google Scholar
  7. [7]
    Natterer, F.: Computerized Tomography with Unknown Sources. To appear in SIAM J. Appl. Math.Google Scholar
  8. [8]
    Peres, A.: Tomographic Reconstruction from Limited Angular Data. J. Comput. Assist. Tomogr. 3, 800–803 (1979).Google Scholar
  9. [9]
    Perry, R.M.: Reconstructing a Function by Circular Harmonic Analysis of its Line Integrals, in: Image Processing for 2D and 3D Reconstruction from Projections, Stanford University, Institute for Electronics in Medicine 1975.Google Scholar
  10. [10]
    Herman, G.T.: Image Reconstruction from Projections. Academic Press 1980.Google Scholar

Copyright information

© Birkhäuser Boston 1983

Authors and Affiliations

  • Frank Natterer

There are no affiliations available

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