Detection and filtering for two-dimensional random fields

In this paper we focus our attention on the following model: Suppose that a two-dimensional random field ??(t1,t2) is observed on a rectangle 0 ?? t1 ?? T1, 0 ?? t2 ?? T2. We assume that the observation consists of a signal x(t1,t2) plus an additive white Gaussian noise, i.e., ??(t1,t2) = x(t1,t2) + ??(t1,t2) where ?? is a white Gaussian noise. We shall consider the following two problems: (a) What is the likelihood ratio as a function of the observation with respect to the situation where ?? is itself a white Gaussian noise? (b) For what models of the signal x will recursive filtering be possible? What is the nature of recursion in this case, and what are the resulting formulas? In this paper we shall summarize the results which have been obtained to date on the two problems of detection and filtering stated above.