  This paper deals with maximum entropy completion of partially specified
block-circulant matrices. Since positive definite symmetric circulants happen
to be covariance matrices of stationary periodic processes, in particular of
stationary reciprocal processes, this problem has applications in signal
processing, in particular to image modeling. In fact it is strictly related to
maximum likelihood estimation of bilateral AR-type representations of acausal
signals subject to certain conditional independence constraints. The maximum
entropy completion problem for block-circulant matrices has recently been
solved by the authors, although leaving open the problem of an efficient
computation of the solution. In this paper, we provide an effcient algorithm
for computing its solution which compares very favourably with existing
algorithms designed for positive definite matrix extension problems. The
proposed algorithm benefits from the analysis of the relationship between our
problem and the band-extension problem for block-Toeplitz matrices also
developed in this paper.
