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 specific scandiction schemes which are universal in some sense.
Specifically, we investigate the following problems: First, modeling the data
array as a random field, we wish to examine whether there exists a scandiction
scheme which is independent of the field's distribution, yet asymptotically
achieves the same performance as if this distribution was known. This question
is answered in the affirmative for the set of all spatially stationary random
fields 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 first part of a two-part paper on sequential decision
making for multi-dimensional data. It deals with clean, noiseless data arrays.
The second part deals with noisy data arrays, namely, with the case where the
decision maker observes only a noisy version of the data, yet it is judged with
respect to the original, clean data.Comment: 46 pages, 2 figures. Revised version: title changed, section 1
revised, section 3.1 added, a few minor/technical corrections mad