  This paper deals with tomographic image reconstruction under the situation
where some of projection data bins are contaminated with abnormal data. Such
situations occur in various instances of tomography. We propose a new
reconstruction algorithm called the Fault-Tolerant reconstruction outlined as
follows. The least-squares (L2-norm) error function ||Ax-b||_2^2 used in
ordinary iterative reconstructions is sensitive to the existence of abnormal
data. The proposed algorithm utilizes the L1-norm error function ||Ax-b||_1^1
instead of the L2-norm, and we develop a row-action-type iterative algorithm
using the proximal splitting framework in convex optimization fields. We also
propose an improved version of the L1-norm reconstruction called the L1-TV
reconstruction, in which a weak Total Variation (TV) penalty is added to the
cost function. Simulation results demonstrate that reconstructed images with
the L2-norm were severely damaged by the effect of abnormal bins, whereas
images with the L1-norm and L1-TV reconstructions were robust to the existence
of abnormal bins.
