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General description of the Problem.
In this paper, we describe how VT framework can be used to reconstruct electric fields induced by focal sources considering Neumann homogeneous boundary conditions. In particular, we study reconstruction of electric fields that are generated by dipole sources which is a case that is often met, for example, in EEG source imaging.
As a comparison to the traditional EEG source imaging problem, we note that VT does not require an explicit mathematical modelling of the sources.To the best of our knowledge, non-zero divergence vector fields in bounded domains have not been previously studied with VT. The previous work related to VT has dealt with divergence-free (or source-free) cases which are mathematically much easier to solve.These previously studied smooth fields have been, for example, velocity fields of moving liquids which can numerically be solved from longitudinal integral measurement using algebraic reconstruction techniques (ART).
In this problem we do not use ART; instead we formulate a convex minimization problem with the help of longitudinal line measurements and sparsity constraints.The constraints were selected based on the physical properties of the problem. In particular, by employing the Radon transform properties, we show how the transverse integral measurements are related to the focal sources and how they can be used as a sparsity constraint even though they cannot be physically measured.
We validate our analysis using simulated data. Our simulations show that the patterns of the electric fields can be correctly estimated and that the location of the source activity can be pointed out from the reconstructed magnitudes of the field.
Publication: A. Koulouri, M. Brookes and V. Rimpilainen. Vector tomography for reconstructing electric Fieldwith non-zero divergence in bounded domains, Journal of Computational Physic,Volume 329, 15 January 2017, Pages 73–90
More details can be found in http://www.sciencedirect.com/science/article/pii/S0021999116305356
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