scholarly journals The method of approximate inverse for the Radon transform operator acting on functions and for the normal Radon transform operators acting on vector and symmetric $2$-tensor fields in $mathbb{R}^3$

2020 ◽  
Vol 17 ◽  
pp. 1073-1087
Author(s):  
I. E. Svetov
Keyword(s):  
Radon Transform ◽  
Tensor Fields ◽  
Approximate Inverse

scholarly journals The Formula of Grangeat for Tensor Fields of Arbitrary Order innDimensions

2007 ◽  
Vol 2007 ◽  
pp. 1-4 ◽  
Author(s):  
T. Schuster
Keyword(s):  
Radon Transform ◽  
Tensor Field ◽  
Arbitrary Order ◽  
Cone Beam ◽  
Inversion Method ◽  
Tensor Fields ◽  
Arbitrary Tensor

The cone beam transform of a tensor field of orderminn≥2dimensions is considered. We prove that the image of a tensor field under this transform is related to a derivative of then-dimensional Radon transform applied to a projection of the tensor field. Actually the relation we show reduces form=0andn=3to 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, form>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.

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