Mathematics in Biomedical ImagingView this Special Issue
Research Article | Open Access
T. Schuster, "The Formula of Grangeat for Tensor Fields of Arbitrary Order in Dimensions", International Journal of Biomedical Imaging, vol. 2007, Article ID 012839, 4 pages, 2007. https://doi.org/10.1155/2007/12839
The Formula of Grangeat for Tensor Fields of Arbitrary Order in Dimensions
The cone beam transform of a tensor field of order in dimensions is considered. We prove that the image of a tensor field under this transform is related to a derivative of the -dimensional Radon transform applied to a projection of the tensor field. Actually the relation we show reduces for and 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 , 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.
- A. K. Louis, “Filter design in three-dimensional cone beam tomography: circular scanning geometry,” Inverse Problems, vol. 19, no. 6, pp. S31–S40, 2003.
- A. Katsevich, “Theoretically exact filtered backprojection-type inversion algorithm for spiral CT,” SIAM Journal on Applied Mathematics, vol. 62, no. 6, pp. 2012–2026, 2002.
- T. Schuster, “An efficient mollifier method for three-dimensional vector tomography: convergence analysis and implementation,” Inverse Problems, vol. 17, no. 4, pp. 739–766, 2001.
- E. Yu. Derevtsov and I. G. Kashina, “Numerical solution to the vector tomography problem by tools of a polynomial basis,” Siberian Journal of Numerical Mathematics, vol. 5, no. 3, pp. 233–254, 2002 (Russian).
- G. Sparr, K. Stråhlén, K. Lindström, and H. W. Persson, “Doppler tomography for vector fields,” Inverse Problems, vol. 11, no. 5, pp. 1051–1061, 1995.
- E. Yu. Derevtsov, “An approach to direct reconstruction of a solenoidal part in vector and tensor tomography problems,” Journal of Inverse and Ill-Posed Problems, vol. 13, no. 3, pp. 213–246, 2005.
- S. G. Kazantsev and A. A. Bukhgeim, “Singular value decomposition for the 2D fan-beam Radon transform of tensor fields,” Journal of Inverse and Ill-Posed Problems, vol. 12, no. 3, pp. 245–278, 2004.
- P. Grangeat, “Mathematical framework of cone beam 3D reconstruction via the first derivative of the Radon transform,” in Mathematical Methods in Tomography, G. T. Herman, A. K. Louis, and F. Natterer, Eds., vol. 1497 of Lecture Notes in Math., pp. 66–97, Springer, Berlin, Germany, 1991.
- V. A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, The Netherlands, 1994.
- F. Natterer, The Mathematics of Computerized Tomography, John Wiley & Sons, Chichester, NH, USA, 1986.
- F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, SIAM, Philadelphia, Pa, USA, 2001.
- A. Denisjuk, “Inversion of the X-ray transform for 3D symmetric tensor fields with sources on a curve,” Inverse Problems, vol. 22, no. 2, pp. 399–411, 2006.
- A. K. Louis, “Approximate inverse for linear and some nonlinear problems,” Inverse Problems, vol. 12, no. 2, pp. 175–190, 1996.
- A. Rieder and T. Schuster, “The approximate inverse in action III: 3D-Doppler tomography,” Numerische Mathematik, vol. 97, no. 2, pp. 353–378, 2004.
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.