Universal Scanning of Mixing Random Fields and the Performance of the Peano-Hilbert Scan

We investigate the problem of scanning and prediction ("scandiction", for short) of multidimensional data arrays. This problem arises in several aspects of image and video processing, such as predictive coding, where an image is compressed by coding the prediction error sequence resulting from scandicting it. Specifically, given a strongly mixing random field, we show that there exists a scandiction scheme which is independent of the field's distribution, yet almost surely asymptotically achieves the same performance as if this distribution was known. We then discuss the scenario where the Peano-Hilbert scanning order is used, accompanied by an optimal predictor, and derive a bound on the excess loss compared to optimal finite state scandiction, which is valid for any individual image and any bounded loss function.