We consider the reconstruction of pictures from p complete projections. We show that the filtered backprojection algorithm achieves an L2-error of order p for picture densities which belong to the Sobolev space \( H_{0}^{\alpha }\left( \Omega \right) \) when the ideal low-pass filter is used. We derive an optimal filter by minimizing our error bound. The validity of our error estimate and the performence of the optimal filter are investigated by numerical experiments.


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  1. [1]
    Budinger, T. F. , Gulberg, G. T. : Three-Dimensional Reconstruction in Nuclear Medicine Emission Imaging, IEEE Transaction on Nuclear Science, 21, 2–20 (1974).CrossRefGoogle Scholar
  2. [2]
    Gordon, R. (ed. ): Image Processing for 2-D and 3-D Reconstruction from Projections: Theory and Practice in Medicine and the Physical Sciences. Optical Society of America, 2000 L Street N. W. , Washington D. C. 20036.Google Scholar
  3. [3]
    Lewitt, R. M. , Bates, R. H. T. , Peters, T. M.: Image Reconstruction From Projections: II: Modified Backprojection Methods, Optik 50, 85–109 (1978).Google Scholar
  4. [4]
    Ludwig, D.: The Radon Transform on Euclidean Space, Comm. Pure Applied Math. 19, 49–81 (1966).CrossRefGoogle Scholar
  5. [5]
    Natterer, F.: A Sobolev Space Analysis of Picture Reconstruction. Technischer Bericht FB 10 der Universität des Saarlandes, 6600 Saarbrücken.Google Scholar
  6. [6]
    Radon, J.: Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Berichte Sächsische Akademie der Wissenschaften 69, 262–279 (1917).Google Scholar
  7. [7]
    Ramachandran, G. N. , Lakshminarayanan, A. V.: Three-dimensional Reconstruction from Radiography and Electron Micrographs: Application of Convolutions Instead of Fourier Transforms, Proc. Nat. Acad. Sci. U. S. 68, 2236–2240 (1971).CrossRefGoogle Scholar
  8. [8]
    Smith, K. T. , Solmon, D. C, Wagner, S. L. : Practical and Mathematical Aspects of the Problem of Reconstructing Objects from Radiographs. Bul. AMS.Google Scholar

Copyright information

© Springer Basel AG 1979

Authors and Affiliations

  • Frank Natterer

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