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

  • O. E. Trofimov
Conference paper

DOI: 10.1007/978-3-540-71992-2_152

Volume 12 of the book series Mathematics in Industry (MATHINDUSTRY)
Cite this paper as:
Trofimov O.E. (2008) Use of the Fourier Transform in the Distributions Sense for Creation Numerical Algorithms for Cone-Beam Tomography. In: Bonilla L.L., Moscoso M., Platero G., Vega J.M. (eds) Progress in Industrial Mathematics at ECMI 2006. Mathematics in Industry (The European Consortium for Mathematics in Industry), vol 12. Springer, Berlin, Heidelberg

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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© Springer-Verlag Berlin Heidelberg 2008

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  • O. E. Trofimov

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