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Talk Abstract

Vector Field Tomography

Vector Field Tomography

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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