Markov pyramid models in image analysis

The use of statistical pattern recognition techniques in image processing has led to simplifying assumptions on the statistical interdependence of the pixel value of an image, which allow theoretical analysis and/or computational implementation to be achieved. For instance, the assumption of statistical independence of the values or that their joint distributions are multivariate normal, simplifies the analysis enormously. However, these results are very limiting in representing models for data, and do not allow for analysis of arbitrary spatial dependencies, in the data. One method for modeling two-dimensional data on a lattice array has been developed by Abend et al. called the Markov mesh model, and is a generalization of the familiar 1D Markov chain. The Markov mesh model allows the use of a class of spatial dependencies that is popular in many 2D data processing schemes, including image processing. One advantage of using this model is that it allows a computationally attractive implementation of statistical procedures involving joint and conditional probabilities. In this paper, we generalize Abend et al.'s results to a more comprehensive model, which we call the Markov pyramid model, using the concept of partial ordering. We present the necessary background for this model and show that Abend's model is a special case of our model. Finally, we present a simple application of our results to texture modeling.