  We propose a novel direct sampling method (DSM) for the effective and stable
inversion of the Radon transform. The DSM is based on a generalization of the
important almost orthogonality property in classical DSMs to fractional order
Sobolev duality products and to a new family of probing functions. The
fractional order duality product proves to be able to greatly enhance the
robustness of the reconstructions in some practically important but severely
ill-posed inverse problems associated with the Radon transform. We present a
detailed analysis to better understand the performance of the new probing and
index functions, which are crucial to stable and effective numerical
reconstructions. The DSM can be computed in a very fast and highly parallel
manner. Numerical experiments are carried out to compare the DSM with a popular
existing method, and to illustrate the efficiency, stability, and accuracy of
the DSM.
