Approximate inverse and Sobolev estimates for the attenuated Radon transform

The ill-posedness of the attenuated Radon transform is a challenging issue in practice due to the Poisson noise and the high level of attenuation. The investigation of the smoothing properties of the underlying operator is essential for developing a stable inversion. In this paper, we consider the framework of Sobolev spaces and derive analytically a reconstruction algorithm based on the method of the approximate inverse. The derived method inherits the efficiency and stability of the approximate inverse and supplies a method of extraction of contours. These algorithms appear to be efficient for an attenuation of human body type. However, for higher attenuations the ill-posedness increases exponentially what deteriorates accordingly the quality of reconstructions. Nevertheless, a high attenuation map affects less the contour extraction of a high contrast function and so can be neglected. This leads to simplifying the proposed method and circumvents in this case the artifacts due to the attenuation as attested by simulation results. (paper)