On the Likelihood Ratio for Two-Parameter Discrete Space Stochastic Processes

In this paper we derive computable expressions for the likelihood ratio for Gaussian processes with a discrete two-dimensional parameter domain (for instance, an image). A general formula for the likelihood ratio can be easily derived, but becomes inefficient if the number of elements in the parameter domain (the number of pixels in the image) is large. Under additional assumptions such as cyclicity and homogeneity, we can derive formulas for the likelihood ratio which can be very easily applied. All results derived in this paper for the two-parameter case can be extended to the multi-parameter case.