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

Abstract

The cone beam transform of a tensor field of order m in n ≥ 2 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.

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

“…We proved a generalization of that formula which is valid for any symmetric, covariant tensor field of rank m in n dimensions in [13] but our presentation will be restricted to D 0 and D 1 .…”

“…It tells that we have full knowledge of R g ( ω , s ) for all ω , s and any scalar function g : Ω 3 → ℝ, if any plane intersecting the object Ω ⊆ Ω 3 does also have at least one intersection point with the scanning curve Γ and this intersection must be nontransversally. This works fine for D 0 since then f ( x ) is independent of α but unfortunately that does not help in case of D 1 (and analogous transforms for tensor fields as well, see [13]), since there the object function f α = 〈 f ( x ), ( x − α )/| x − α |〉 ℝ 3 of R depends on and hence changes with α , see (15). Thus we seek an alternative way of solving D 1 f = g for vector fields f .…”

“…One of the crucial tools when computing reconstruction kernels in scalar cone beam tomography is the formula of Grangeat [ 14 ]. We proved a generalization of that formula which is valid for any symmetric, covariant tensor field of rank m in n dimensions in [ 13 ] but our presentation will be restricted to D 0 and D 1 .…”

“…First we state essential mathematical properties of ( 1 ) for the special cases of m = 0 and m = 1. The interested reader will find generalizations of the presented theorems to symmetric, covariant tensor fields of any rank and in any dimension in [ 13 ], especially the extension of Grangeat's formula which has been proven for ( 1 ) in case n = 3, m = 0 (see ( 2 )) in [ 14 ]. Then we outline how the method of approximate inverse can be used for ( 2 ) to solve D 0 f = g and present its application to ( 3 ) to solve D 1 f = g .…”

An exact inversion formula for cone beam vector tomography

“…The question thus arose whether any of these formulas could be extended to the case of solenoidal vector fields. Schuster [Sch07] proved an extension of the formula of Grangeat for vector and even tensor fields of arbitrary rank. However, this formula could not be used to derive an exact inversion formula similar to how it was done for scalar fields in [Gra91].…”