On the range of the attenuated Radon transform in strictly convex sets

Abstract

We present new necessary and sufficient conditions for a function on@×S 1 to be in the range of the attenuated Radon transform of a sufficiently smooth function support in the convex set ⊂ 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. In this paper we are concerned with the range characterization of the attenuated Radon transform of function of compact support in the plane. Necessary and sufficient constraints on range of the non-att enuated (clas- sical) Radon transform in the Euclidean space have been known since the works in (6), (7), and (10). These constraints, known as the Cavalieri or the moment conditions, are in terms of the angular variable. For function in the Schwartz class, they are essentially unique due to a Paley-Wiener type the- orem. Moreover, the Helgason support theorem extends the conditions to smooth functions of compact support (8). However, in the case of functions of compact support, it is possible to obtain essentially dif ferent range con- ditions since more than one operator can annihilate functions of compact support in the range of the Radon transform. The results here constitute one such example. Inversion methods of the attenuated Radon transform in the plane ap- peared first in (1), and (14), and various developments can be found in (13), (3), (5), (2). The interest in range conditions stems out fro m their appli- cations to data enhancement in medical imaging methods such as Single Photon, or Positron Emission Computed Tomography (12). For the Eu- clidean attenuated Radon transform, some range characterization based on the inversion procedure in (14) can be found in (15). These constraints are also in terms of the angular variable. Different from the existing results above, our new characterization is in terms of a Hilbert transform associated with the A-analytic maps ` a la