Use of the Fourier Transform in the Distributions Sense for Creation Numerical Algorithms for Cone-Beam Tomography

  • O. E. Trofimov
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 12)

Let the homogeneous of L degree function g(x) be defined in Ndimensional space, and let the function G(y) be its Fourier transform in the distribution sense. The theorem that allows to present the function G(y) using only the values of function g(x) on the unit sphere is proved in the chapter for the case L > −N. The case N=3 and L = −1 corresponds to the properties of beam transform in 3D space. In the chapter it is shown how the theorem may be used for creation of numerical algorithms for cone-beam tomography.

Let the homogeneous of L degree function g(x) be defined in N-dimensional space, and let the function G(y) be its Fourier transform. In view of homogeneity, the function g(x) and its Fourier transform in sense of distributions are defined by their values on the unit sphere [GS00]. We will prove the theorem that allows to present the function G(y) using only the value of function g(x) on the unit sphere for the case L > −N.

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References

  1. [GS00]
    Gelfand, I.M. and Shilov, G.E. Distributions, v.1, Distributions and Actions over them. Moscow.2000.Google Scholar
  2. [LZT01]
    Lavrentiev M.M., Zerkal S.M., Trofimov O.E. Computer Modelling in Tomography and Ill-Posed Problems, VSP (The Netherlands), 2001, 128 pp.Google Scholar
  3. [Nat86]
    F. Natterer, The Mathematics of Computerized Tomography. B.G. Teubner, Stutgart, and John Willey & Sons Ltd, 1986.Google Scholar
  4. [Tr04]
    Trofimov O.E. On Fourier Transform of a Class of Generalized Homogeneous Functions. Siberian Journal of Industrial Mathematics, 2004. Vol. 7, No. 1. pp. 30-34. (in Russian)MathSciNetGoogle Scholar
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    Tuy H. K., An Inversion Formula for Cone-beam Reconstruction. SIAM. J. APPL. MATH. 1983, vol. 43, N 3, PP. 546-552.CrossRefMathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2008

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  • O. E. Trofimov

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