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 $\mathbb{R}^3$

I. E. Svetov

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1073-1087.

Résumé

We propose approach for reconstruction of a three-dimensional function from the known values of Radon transform. The approach is based on the method of approximate inverse. The obtained result is the basis of two approaches for reconstruction of a potential part of vector and symmetric $2$-tensor fields, which have form $\mathrm{d}\psi$, $\psi\in H^1_0(B)$ and $\mathrm{d}^2\psi$, $\psi\in H^2_0(B)$, respectively. Here $\mathrm{d}$ is the inner derivation operator, which is a composition of the operators of gradient and symmetrization. Initial data for the problems are the known values of normal Radon transform. The first approach allows to recover components of potential part of fields, and the second reconstructs a potential of potential part of fields.

Détail

Keywords: tensor tomography, method of approximate inverse, adjoint operator, vector field, symmetric $2$-tensor field, potential field, potential.
Mots-clés : Radon transform, normal Radon transform

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