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Mathematical framework of cone beam 3D reconstruction via the first derivative of the radon transform

  • Medical Imaging Techniques
  • Conference paper
  • First Online:
Mathematical Methods in Tomography

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1497))

Abstract

Either for medical imaging or for non destructive testing, X-ray provides a very accurate mean to investigate internal structures. The object is described by a 3D map f of the local density. The use of a 2D X-ray detector like an image intensifier in front of the ponctual X-ray source defines a cone beam geometry. When the source moves along a curve, the acquisition measurements are modelized by the cone beam X-ray transform of the function f. This same model can be applied to emission tomography when cone beam collimators are used.

The aim of the 3D reconstruction is to recover the original function f. We have established an exact formula between the cone beam X-ray transform and the first derivative of the 3D Radon transform. We propose to use the planes as information vectors to achieve the rebinning from the coordinates system linked to the cone beam geometry, to the spherical coordinates system of the Radon domain. Then the reconstruction diagram is to compute and to invert the first derivative of the 3D Radon transform.

In this publication, we describe the mathematical framework of this reconstruction diagram. We emphasize the special case of the circular acquisition trajectory.

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Author information

Authors and Affiliations

  1. LETI - Département Systèmes - SETIA, Centre d'Etudes Nucléaires de Grenoble, 85 X - 38041, Grenoble Cedex, France

    Pierre Grangeat

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  1. Pierre Grangeat
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Editor information

Editors and Affiliations

  1. Department of Radiology, University of Pennsylvania, Philadelphia, PA, 19104, USA

    Gabor T. Herman

  2. Fachbereich Mathematik, Universität des Saarlandes, 6600, Saarbrücken, Germany

    Alfred K. Louis

  3. Institut für Numerische und Instrumentelle Mathematik, Universität Münster, Einsteinstraße 62, 4400, Münster, Germany

    Frank Natterer

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© 1991 Springer-Verlag Berlin Heidelberg

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Grangeat, P. (1991). Mathematical framework of cone beam 3D reconstruction via the first derivative of the radon transform. In: Herman, G.T., Louis, A.K., Natterer, F. (eds) Mathematical Methods in Tomography. Lecture Notes in Mathematics, vol 1497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084509

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  • DOI: https://doi.org/10.1007/BFb0084509

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54970-3

  • Online ISBN: 978-3-540-46615-4

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