This paper introduces a nonparametric algorithm for bootstrapping a stationary random field and proves certain consistency properties of the algorithm for the case of mixing random fields. The motivation for this paper comes from relating a heuristic texture synthesis algorithm popular in computer vision to general nonparametric bootstrapping of stationary random fields. We give a formal resampling scheme for the heuristic texture algorithm and prove that it produces a consistent estimate of the joint distribution of pixels in a window of certain size under mixing and regularity conditions on the random field. The joint distribution of pixels is the quantity of interest here because theories of human perception of texture suggest that two textures with the same joint distribution of pixel values in a suitably chosen window will appear similar to a human. Thus we provide theoretical justification for an algorithm that has already been very successful in practice, and suggest an explanation for its perceptually good results.
"Texture synthesis and nonparametric resampling of random fields." Ann. Statist. 34 (4) 1751 - 1773, August 2006. https://doi.org/10.1214/009053606000000588