Abstract
In this paper, we introduce a notion of spatial redundancy in Gaussian random fields. This study is motivated by applications of thea contrariomethod 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 squaredL2norm, 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.
Highlights
Stochastic geometry [3, 12, 52] aims at describing the arrangement of random structures based on the knowledge of the distribution of geometrical elementary patterns
When the considered patterns are functions over some topological space, we can study the geometry of the associated random field
Centering a kernel function at each point of a Poisson point process gives rise to the notion of shot-noise random field [18, 50, 51]
Summary
Stochastic geometry [3, 12, 52] aims at describing the arrangement of random structures based on the knowledge of the distribution of geometrical elementary patterns (point processes, random closed sets, etc.). We derive explicit probability distribution functions for the random variables associated with the output of similarity functions computed on local windows of random fields. The knowledge of such functions allows us to conduct rigorous statistical testing on the spatial redundancy in natural images. This assumption will allow us to explicitly derive moments of some similarity functions computed on local windows of the random field Once again, another reason for this restriction comes from image processing. Technical proofs and additional results on multidimensional central limit theorems are presented in the Appendices
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