  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.
