Scanning, filtering and prediction for random fields corrupted by Gaussian noise

We consider the problem of sequential decision making on random fields corrupted by Additive White Gaussian Noise (AWGN). In particular, we first consider the problem of sequentially filtering an AWGN-corrupted random field. In this scenario, the sequential filter may be given the freedom to choose the path over which it traverses the random field (e.g., noisy image), thus it is natural to ask what is the best achievable performance and how far is the performance of widely used scanning methods from the optimum. We formally define the problem of scanning and filtering, derive a bound on the best achievable performance and quantify the excess loss occurring when non-optimal scanners are used, compared to optimal scanning and filtering. We then discuss the problem of sequential scanning and prediction of noisy random fields. This setting is a natural model for applications such as restoration and coding of noisy images. In this scenario, using predictive coding methods on the noisy image results in both enhancement and compression of the input image, as one expects that the prediction error consists mainly of the noise signal. We formally define the problem of sequential prediction in a noisy array and compute the optimal performance in terms of the clean scandictability defined by Merhav and Weissman.