  The modern imaging techniques of Positron Emission Tomography and of Single
Photon Emission Computed Tomography are not only two of the most important
tools for studying the functional characteristics of the brain, but they now
also play a vital role in several areas of clinical medicine, including
neurology, oncology and cardiology. The basic mathematical problems associated
with these techniques are the construction of the inverse of the Radon
transform and of the inverse of the so called attenuated Radon transform
respectively. We first show that, by employing mathematical techniques
developed in the theory of nonlinear integrable equations, it is possible to
obtain analytic formulas for these two inverse transforms. We then present
algorithms for the numerical implementation of these analytic formulas, based
on approximating the given data in terms of cubic splines. Several numerical
tests are presented which suggest that our algorithms are capable of producing
accurate reconstruction for realistic phantoms such as the well known
Shepp--Logan phantom.
