ABSTRACT. We present new necessary and sufficient conditions for a function on ∂Ω×S 1 to be in the range of the attenuated Radon transform of a sufficiently smooth function support in the convex set Ω ⊂ R 2 . The approach is based on an explicit Hilbert transform associated with traces of the boundary of A-analytic functions in the sense of Bukhgeim.

Abstract. We characterize the range of the attenuated and non-attenuated X-ray transform of compactly supported vector fields in the plane. The characterization is in terms of a Hilbert transform associated with the A-analytic functionsà la Bukhgeim. As an application we determine necessary and sufficient conditions for the attenuated Doppler and X-ray data to be mistaken for each other. The problem of inversion of the X-ray transform of higher order tensor fields have been formulated in [30] in the geometric setting of Riemannian manifolds with boundary; see [33] for the Euclidean setting. Coming from the practical procedure of acquiring data, the X-ray transform of vector fields is also known as the Doppler transform. Various partial results (e.g., [31], [32]) culminated with the inversion formulas for recovering the solenoidal part of one-tensors on simple Riemannian surfaces with boundary [27]; see also [28]. Injectivity in the attenuated case for both 0-and 1-tensors is much more recent [34]; see also [12] for a more general weighted transform. Inversion formulas to tensors of higher orders have been found in [15] for the Euclidean case, and [25] for the Riemannian case. However, these works do not address range characterization.The first range characterizations of the (non-attenuated) X-ray transform is given in [27] for both 0-and 1-tensors supported on simple Riemannian surfaces with boundary. This characterization is given in terms of the scattering relation.We consider here the problem of the range characterization of both attenuated and non-attenuated Doppler transform in a strictly convex bounded domain in the Euclidian plane. Our approach uses the Hilbert transform for A-analytic maps in [29] and relies on new identities enjoyed by such maps, see Lemma 2.6. The characterization concerns the Fourier modes (in the angular variables) of the data and can be

In this paper we study the attenuated X-ray transform of 2-tensors supported in strictly convex bounded subsets in the Euclidean plane. We characterize its range and reconstruct all possible 2-tensors yielding identical X-ray data. The characterization is in terms of a Hilbert-transform associated with A-analytic maps in the sense of Bukhgeim.

We revisit the inverse source problem in a two dimensional absorbing and scattering medium and present a non-iterative reconstruction method using measurements of the radiating flux at the boundary. The attenuation and scattering coefficients are known and the unknown source is isotropic. The approach is based on the Cauchy problem for a Beltrami-like equation for the sequence valued maps, and extends the original ideas of A. Bukhgeim from the non-scattering to scattering media. We demonstrate the feasibility of the method in a numerical experiment in which the scattering is modeled by the two dimensional Henyey-Greenstein kernel with parameters meaningful in Optical Tomography.