In this paper, we introduce a notion of spatial redundancy in Gaussian random fields. This study is motivated by applications of the a contrario method in image processing. We define similarity functions on local windows in random fields over discrete or continuous domains. We derive explicit Gaussian asymptotics for the distribution of similarity functions when computed on Gaussian random fields. Moreover, for the special case of the squared L2 norm, we give non-asymptotic expressions in both discrete and continuous periodic settings. Finally, we present fast and accurate approximations of these non-asymptotic expressions using moment methods and matrix projections.