Discrete-index Markov-type random processes

Abstract

Discrete-index Markov-type random processes in one and two dimensions are considered, with emphasis on two-dimensional processes (or fields). Important classes of Markov-type models, their properties, and their relationship are described. Although some new results are given, the authors mainly present a systematic study and grouping of processes according to two fundamental Markov-type properties: strict-sense Markov, defined in terms of conditional probabilities, and wide-sense Markov, defined in terms of linear minimum-mean-square error estimates. Classes of models having special cases of the fundamental properties, including many models which are widely used to represent images are obtained by specifying the index set, the conditioning set used to define the Markov property, and the process distribution. The relationships between unilateral and bilateral models in each class are carefully investigated. Particular attention is given to simultaneous autoregressive models which are shown to be both strict-sense and wide-sense Markov. Classification of processes according to their Markov-type properties helps to clarify the consequences of and relationships between different model assumptions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>