Relations between the likelihood ratios for 2D continuous and discrete time stochastic processes

The author considers the likelihood ratio for 2D processes. In order to detect this ratio, it is necessary to compute the determinant of the covariance operator of the signal-plus-noise observation process. In the continuous case, this is in general a difficult problem. For cyclic processes, using Fourier transforms it is possible to compute the determinant for continuous and discrete processes. For the 2D Poisson equation and its discretization, it is shown that the discretized determinant converges to the continuous one if the stepsize tends to zero