Unique tomographic reconstruction of vector fields using boundary data

In tomography a 2-D scalar function f (r) is reconstructed from its line integrals through some domain D of f. The related problem of reconstructing a vector field upsilon(r) from its line integrals (through D) is generally underdetermined since upsilon(r) is defined by two component functions. When upsilon(r) is decomposed into its irrotational and solenoidal components, upsilon = del Phi + del x Psi, it is shown that the solenoidal part del x Psi is uniquely determined by the line integrals of upsilon(r). This is demonstrated here in a particularly simple manner in the Fourier domain using a vector analog of the well-known projection slice theorem. In addition, under the constraint that upsilon(r) is divergenceless in D, a formula for the scalar potential Phi(r) is given in terms of the normal component of upsilon(r) on the boundary of D. Thus when no sources or sinks reside in D, implying upsilon(r) is divergenceless there, two types of data uniquely determine the vector field upsilon(r) in D: the line integrals of upsilon(r) through D (tomographic data) and the normal component of upsilon(r) on the boundary D. An important application of vector tomography arises in the problem of mapping a fluid velocity field from reciprocal acoustic travel time measurements or Doppler backscattering measurements.