  This is a survey of some recent results on the rational circulant covariance
extension problem: Given a partial sequence $(c_0,c_1,\dots,c_n)$ of covariance
lags $c_k=\mathbb{E}\{y(t+k)\overline{y(t)}\}$ emanating from a stationary
periodic process $\{y(t)\}$ with period $2N>2n$, find all possible rational
spectral functions of $\{y(t)\}$ of degree at most $2n$ or, equivalently, all
bilateral and unilateral ARMA models of order at most $n$, having this partial
covariance sequence. Each representation is obtained as the solution of a pair
of dual convex optimization problems. This theory is then reformulated in terms
of circulant matrices and the connections to reciprocal processes and the
covariance selection problem is explained. Next it is shown how the theory can
be extended to the multivariate case. Finally, an application to image
processing is presented.
