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Vector tomography methods intend to reconstruct and visualize vector fieldsin restricted domains by measuring line integrals of projections of thesevector fields. Here, we deal with the reconstruction of irrotational vectorfunctions from boundary measurements. As the majority of inverse problems,vector field recovery is an ill posed in the continuous domain and thereforefurther assumptions, measurements and constraints should be imposed for thefull vector field estimation. The reconstruction idea in the discrete domainrelies on solving a numerical system of linear equations which derives from theapproximation of the line integrals along lines which trace the bounded domain.This work presents an extensive description of a vector field recovery, thefundamental assumptions and the ill conditioning of this inverse problem. Moreimportantly we show that this inverse problem is regularized via the domaindiscretization, i.e. we show that the recovery of an irrotational vector fieldwithin a discrete grid employing a finite set of longitudinal line integrals,leads to a consistent linear system which has bounded solution errors. Weelaborate on the estimation of the solution's error and we prove that thisrelative error is finite and therefore a stable vector field reconstruction isensured. Such theoretical aspects are critical for future implementations ofvector tomography in practical applications like the inverse bioelectric fieldproblem. We validate our theoretical results by performing simulations thatreconstruct smooth irrotational fields based solely on a finite number ofboundary measurements and without the need of any additional or priorinformation (e.g. transversal line integrals or source free assumption).

Overcoming the ill-posedness through discretization in vector tomography: Reconstruction of irrotational vector fields