Giorgio Picci

Statistical modeling of images has been the subject of intense
research in the past two decades and forms now a vast literature. Most
of the literature deals with so-called *Gibbs-Markov* models for
random fields borrowed (with some smart twists) from statistical
mechanics. Unfortunately these models lead to complicated estimation
problems which have to be approached by Monte-Carlo type techniques,
such as simulated annealing, MCMC, etc.

We instead propose to use a simple class of stochastic models, known
as *reciprocal processes*. These can actually be seen as a
special class of G-M random fields and have been well studied in 1-D,
especially by Arthur Krener and his collaborators. It can in
particular be shown that stationary reciprocal processes admit a
*descriptor type* representation of a certain kind which can be seen
as a natural non-causal extension of the popular linear state space
models used in control and time series analysis. One should stress
that reciprocal processes, in particular stationary reciprocal
processes, naturally live in a finite region of the "time" line (or of the
plane) and the descriptor models are associated with certain boundary
conditions. Estimation and identification of certain classes of
these models can in principle be rephrased as a classical problem
of *banded extension* for Toeplitz and block-circulant matrices.