  PURPOSE: We develop a practical, iterative algorithm for image-reconstruction
in under-sampled tomographic systems, such as digital breast tomosynthesis
(DBT).
  METHOD: The algorithm controls image regularity by minimizing the image total
$p$-variation (TpV), a function that reduces to the total variation when
$p=1.0$ or the image roughness when $p=2.0$. Constraints on the image, such as
image positivity and estimated projection-data tolerance, are enforced by
projection onto convex sets (POCS). The fact that the tomographic system is
under-sampled translates to the mathematical property that many widely varied
resultant volumes may correspond to a given data tolerance. Thus the
application of image regularity serves two purposes: (1) reduction of the
number of resultant volumes out of those allowed by fixing the data tolerance,
finding the minimum image TpV for fixed data tolerance, and (2) traditional
regularization, sacrificing data fidelity for higher image regularity. The
present algorithm allows for this dual role of image regularity in
under-sampled tomography.
  RESULTS: The proposed image-reconstruction algorithm is applied to three
clinical DBT data sets. The DBT cases include one with microcalcifications and
two with masses.
  CONCLUSION: Results indicate that there may be a substantial advantage in
using the present image-reconstruction algorithm for microcalcification
imaging.
