Abstract
We consider the reconstruction of pictures from p complete projections. We show that the filtered backprojection algorithm achieves an L2-error of order p-α 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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Literatur
Budinger, T. F. , Gulberg, G. T. : Three-Dimensional Reconstruction in Nuclear Medicine Emission Imaging, IEEE Transaction on Nuclear Science, 21, 2–20 (1974).
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.
Lewitt, R. M. , Bates, R. H. T. , Peters, T. M.: Image Reconstruction From Projections: II: Modified Backprojection Methods, Optik 50, 85–109 (1978).
Ludwig, D.: The Radon Transform on Euclidean Space, Comm. Pure Applied Math. 19, 49–81 (1966).
Natterer, F.: A Sobolev Space Analysis of Picture Reconstruction. Technischer Bericht FB 10 der Universität des Saarlandes, 6600 Saarbrücken.
Radon, J.: Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Berichte Sächsische Akademie der Wissenschaften 69, 262–279 (1917).
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).
Smith, K. T. , Solmon, D. C, Wagner, S. L. : Practical and Mathematical Aspects of the Problem of Reconstructing Objects from Radiographs. Bul. AMS.
Editor information
Rights and permissions
Copyright information
© 1979 Springer Basel AG
About this chapter
Cite this chapter
Natterer, F. (1979). Genauigkeitsfragen bei der Numerischen Rekonstruktion von Bildern. In: Ansorge, R., Glashoff, K., Werner, B. (eds) Numerical Mathematics / Numerische Mathematik. ISNM International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale D’Analyse Numérique, vol 49. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6285-1_10
Download citation
DOI: https://doi.org/10.1007/978-3-0348-6285-1_10
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-6286-8
Online ISBN: 978-3-0348-6285-1
eBook Packages: Springer Book Archive