An efficient algorithm for maximum entropy extension of block-circulant covariance matrices

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 efficient algorithm for computing its solution which compares very favorably 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.