  To reduce the x-ray dose in computerized tomography (CT), many constrained
optimization approaches have been proposed aiming at minimizing a regularizing
function that measures lack of consistency with some prior knowledge about the
object that is being imaged, subject to a (predetermined) level of consistency
with the detected attenuation of x-rays. Proponents of the shearlet transform
in the regularizing function claim that the reconstructions so obtained are
better than those produced using TV for texture preservation (but may be worse
for noise reduction). In this paper we report results related to this claim. In
our reported experiments using simulated CT data collection of the head,
reconstructions whose shearlet transform has a small $\ell_1$-norm are not more
efficacious than reconstructions that have a small TV value. Our experiments
for making such comparisons use the recently-developed superiorization
methodology for both regularizing functions. Superiorization is an automated
procedure for turning an iterative algorithm for producing images that satisfy
a primary criterion (such as consistency with the observed measurements) into
its superiorized version that will produce results that, according to the
primary criterion are as good as those produced by the original algorithm, but
in addition are superior to them according to a secondary (regularizing)
criterion. The method presented for superiorization involving the $\ell_1$-norm
of the shearlet transform is novel and is quite general: It can be used for any
regularizing function that is defined as the $\ell_1$-norm of a transform
specified by the application of a matrix. Because in the previous literature
the split Bregman algorithm is used for similar purposes, a section is included
comparing the results of the superiorization algorithm with the split Bregman
algorithm.
