3rd International Workshop on Linear Systems (LinSys2007)

Modeling of images by reciprocal processes
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.