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International Journal of Biomedical Imaging
Volume 2007, Article ID 12839, 4 pages
Research Article

The Formula of Grangeat for Tensor Fields of Arbitrary Order in n Dimensions

Department of Mechanical Engineering, Helmut Schmidt University, P.O. Box 700822, Hamburg 22008, Germany

Received 18 October 2006; Revised 7 February 2007; Accepted 7 February 2007

Academic Editor: Alfred Karl Louis

Copyright © 2007 T. Schuster. 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.


The cone beam transform of a tensor field of order m in n2 dimensions is considered. We prove that the image of a tensor field under this transform is related to a derivative of the n-dimensional Radon transform applied to a projection of the tensor field. Actually the relation we show reduces for m=0 and n=3 to the well-known formula of Grangeat. In that sense, the paper contains a generalization of Grangeat's formula to arbitrary tensor fields in any dimension. We further briefly explain the importance of that formula for the problem of tensor field tomography. Unfortunately, for m>0, an inversion method cannot be derived immediately. Thus, we point out the possibility to calculate reconstruction kernels for the cone beam transform using Grangeat's formula.