Universal Scanning and Sequential Decision Making for Multidimensional Data

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, for example, where an image is compressed by coding the error sequence resulting from scandicting it. Thus, it is natural to ask what is the optimal method to scan and predict a given image, what is the resulting minimum prediction loss, and whether there exist speciflc scandiction schemes which are universal in some sense. Speciflcally, we investigate the following problems: First, modeling the data array as a random fleld, we wish to examine whether there exists a scandiction scheme which is independent of the fleld’s distribution, yet asymptotically achieves the same performance as if this distribution was known. This question is answered in the a‐rmative for the set of all spatially stationary random flelds and under mild conditions on the loss function. We then discuss the scenario where a non-optimal scanning order is used, yet accompanied by an optimal predictor, and derive bounds on the excess loss compared to optimal scanning and prediction. This paper is the flrst part of a two-part paper on sequential decision making for multidimensional data. It deals with clean, noiseless data arrays. The second part [1] deals with