Reconstructions from Doppler Radon transforms

We will study whether it is possible to regain information about a vector eld from projections, in the same way as is possible for scalar functions with Radon transform. It turns out that the curl and divergence of the eld can be reconstructed separately from two different transforms, and algorithms are devised using a variant of the ltered back projection algorithm for reconstruction. It is a well known fact that one can reconstruct a smooth scalar function from all its line integrals (inversion of the Radon transform). There are several diie-rent methods to solve this problem numerically when only a nite number of line integrals are known. In this paper we study whether it is possible to reconstruct a vector valued function from tomographic measurements in a similar way. This problem has also been studied by among others Braun and Hauk 1], Norton 2] and, in a very general setting, Sharafutdinov 3]. The origin of the quantity mainly studied in this paper is Doppler measurements on ows with continuous ultrasound, which has the advantage over pulsed ult-rasound of higher sensitivity, when the velocities are small.