Iterative inversion of the tensor momentum x-ray transform

Abstract

The momentum x-ray transform maps a symmetric tensor field f of order m in to its integrals over lines at + b with the weight tk , . In this article we propose an iterative approach to the reconstruction problem. Namely, a new concept of partial momentum transforms is introduced. These transforms are intermediaries between the momentum and the component-wise x-ray transforms. Moreover, one can use a differential operator to construct a sequence of partial momentum transforms that starts with the momentum and finishes with the component-wise transform. The latter can be inverted with the standard component-wise backprojection. The proposed method was also applied to obtain an analog of the Plancherel formula as well as the stability estimates in the Sobolev spaces that are valid for any m.