Some Integral Transforms of Vector Fields

In the inversion of the Radon and X-ray transforms, a scalar function f is found from its integrals over hyperplanes and lines, respectively. In this paper we demonstrate how to deøne dioeerent kinds of Radon transforms and X-ray transforms for vector øelds in a natural way. It is shown that having data from one such Radon and one such X-ray transform it is possible to reconstruct the vector øeld uniquely in R n , n 2. Here, in a Helmholtz decomposition of the vector øeld, the solenoid part is obtained from an X-ray transform, and the potential part from a Radon transform. In two dimensions it is shown that this approach even works for the exponential counterpart of the Radon and X-ray transforms, but then each of the solenoid and potential parts appear for each of the transforms. Inversion formulas similar to the widely used øltered back projection formula are deviced for these vectorial transforms. The motivation to study these kinds of problems has been ultrasound measurements of..