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International Journal of Biomedical Imaging
Volume 2008, Article ID 174283, 17 pages
http://dx.doi.org/10.1155/2008/174283
Research Article

A Reconstruction Approach for Imaging in 3D Cone Beam Vector Field Tomography

1Department of Mechanical Engineering, Helmut Schmidt University, Holstenhofweg 85, 22043 Hamburg, Germany
2Department of Applied Mathematics, Saarland University, P.O. Box 15 11 50, 66041 Saarbrücken, Germany

Received 4 July 2008; Accepted 15 September 2008

Academic Editor: J. C. Chen

Copyright © 2008 T. Schuster et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

3D cone beam vector field tomography (VFT) aims for reconstructing and visualizing the velocity field of a moving fluid by measuring line integrals of projections of the vector field. The data are obtained by ultrasound measurements along a scanning curve which surrounds the object. From a mathematical point of view, we have to deal with the inversion of the vectorial cone beam transform. Since the vectorial cone beam transform of any gradient vector field with compact support is identically equal to zero, we can only hope to reconstruct the solenoidal part of an arbitrary vector field. In this paper we will at first summarize important properties of the cone beam transform for three-dimensional solenoidal vector fields and then propose a solution approach based on the method of approximate inverse. In this context, we intensively make use of results from scalar 3D computerized tomography. The findings presented in the paper will continuously be illustrated by pictures from first numerical experiments done with exact, simulated data.