Abstract: The ray transform $I$ (also called the Doppler transform) measures the work of a vector field over lines. The operator $I$ has a nontrivial kernel, only the solenoidal part of a vector field $f$ can be recovered from $If$. In the two-dimensional case, we derive an analogoue of the Cormack inversion formula which recovers a solenoidal vector field from integrals measured over lines that do not intersect a certain disk. Then we study the exterior problem for the two-dimensional Doppler transform in two cases: (1) for vector fields defined in a bounded annulus and (2) for vector fields in an unbounded annulus. The theorem on decomposition of a vector field into solenoidal and potential parts is proved in both cases. These two theorems are very different; in particular, the decomposition is not unique in the case of an unbounded annulus. The algorithm of recovering the solenoidal part of a vector field is presented in both cases. Finally a numerical example of reconstructing a solenoidal vector field is presented.

Cite: Sharafutdinov V.A. , Vaitsel N.A.
The Cormack inversion formula for Doppler tomography in two dimendions
Quasilinear Equations, Inverse Problems and their Applications 2025 06-10 Oct 2025