The problem considered is to reconstruct a vector eld in the case when for each line the distribution of the vector eld components along the line is known. Such data are obtained from spectral analysis of signals, a ected by Doppler shifts, caused by re ections on moving particles (\velocity spectra"). In the general setting, the problem is unsolved. Using only integrated data along lines, one encounters the same problem as when reconstructing vector elds by means of time-of-ight measurements. Reconstruction formulas are given in this case, establishing the known fact that only the curl of the vector eld can be reconstructed without additional information. Inspired by the application to which the paper is most devoted, ultrasound Doppler measurements on blood ows, the case where the ow takes place in narrow channels is investigated in simulations and experiments.

No abstract

In the exponential Radon transform in 1~2, the integrals of a scalar function f over lines, with exponential weight functions, are determined. In this paper we demonstrate how to define two different kinds of exponential Radon transforms for vector fields in R 2 in a natural way. It is shown that having data from these transforms it is possible to reconstruct the vector field uniquely. The motivation to study this problem is ultrasound measurements of flows, from which velocity spectra along lines can be determined. The first moment of these can be interpreted by means of one of the exponential vectorial Radon transforms.

No abstract

No abstract