Computerized tomography usually aims at the reconstruction of some scalar density function from data collected along lines. Analogous problems for reconstruction of vector fields have recently been studied by several authors, motivated by different applications. An overview of this work will be given in the talk.

In many situations, e.g. time of flight and Doppler measurements on flows, the integral of the component of the flow vector along lines can be measured. However, from this information, it is only possible to compute the curl of the vector field. The remaining part of the field, the divergence, can in principle be determined from analogous information about the normal of the vector field, a kind of data which however is difficult to acquire in practice. Using apriori-information about the field, e.g. that it is bound to `vessels', where it has a parabolic velocity profile, uniqueness can be achieved, without using the normal components.

It is natural to consider also generalized transforms, modeling exponential decay along the lines of measurement. Also in this case it is possible to formulate reconstruction methods.

A problem of a different nature, much more difficult, containing also non-linear, combinatorial ingredients, is reconstruction in the case when the distribution of the vector field components along lines are known, i.e. `velocity spectra'. This situation occurs naturally in connection with ultrasound Doppler measurements. Some results about this problem will be discussed, using a parametrization that also turns out to be useful in scalar tomography.

Applications can be found in ultrasonic imaging in medicine, flow imaging in non-destructive testing, and in photoelasticity. Results of some experiments and simulations of ultrasound Doppler measurements will be reported.

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1996-1997 Mathematics in High Performance Computing