Deriving deterministic prediction rules from reduction schemes

Abstract

Deterministic Prediction in progressive coding of images is investigated. Progressive coding first creates a sequence of resolution layers by beginning with an original image and reducing its resolution several times by factors of two. Next, the resultant layers are losslessly encoded. The lowest-resolution layer is encoded first, then each higher resolution image is built incrementally upon the previous, until the original image is finally encoded. Coding efficiency may be improved if knowledge of the rules which produced the lower-resolution image of each pair is used to deterministically predict pixels of the higher, so they need not be encoded.

We address this problem: given reduction rules expressing each low-resolution pixel as a function of nearby high-resolution pixels and previously-generated low-resolution pixels, find a complete set of rules, each of which deterministically predicts the value of a high-resolution pixel when certain values are found in nearby low-resolution pixels and previously-coded high-resolution pixels. We show that this problem is NP-Hard by analogy to the well-known Satisfiability problem, then propose a recursive algorithm for solving it in optimal time as a depth-first tree search. The effectiveness of prediction is shown to vary depending on the sequence order in which the pixels are processed; we prove upper bounds on the effectiveness and demonstrate how to find the optimal order. Reduction rules taking their inputs from an area of pixels (template) larger than 2 × 2 are shown to exhibit an interdependence between their input pixels such that certain combinations are impossible; we show a sequence order of processing that addresses correctly this phenomenon. Finally, we demonstrate how multiple template positions lead to a recursion in prediction which further enhances predictability. This work serves as a basis for both working software implementation and for future research into unsolved problems which we identify.