Reconstruction of Scalar and Vector Fields from Their Attenuated Radon Transform

In this paper we derive explicit inversion formulas for recovering scalar and vector fields supported on the unit disk from their 2D attenuated Radon transform in the framework of the fan-beam scanning geometry. The attenuation function is supposed to be known (from additional transmission measurements, for instance). It is known that in the vector case the kernel of the (unattenuated) Radon transform is non-zero and it’s possible to reconstruct only the solenoidal part of the vector field from it. The full reconstruction requires additional information. However, in the presence of a non-zero attenuation we show that knowing the attenuated Radon transform of the vector field is sufficient for reconstructing the whole vector field. Numerical examples are presented.