Causal Markov Random Fields

Abstract

Traditionally two groups have developed extensions of 1-D Markov processes to 2-D image data. People in the first group adopt most of their ideas and tools from statistical mechanics and express the Markov nature of a random field in a noncausal way. The MRFs described in Chapter 2 are such models. The primary goal of the second group is to extend 1-D hidden Markov models (HMMs) to 2-D causal MRF models. The chief obstacle to this extension is the lack of a natural ordering for a two dimensional grid and hence the lack of a natural notion of causality in the spatial image data. As a result, an artificial ordering for image data must be assumed, which sometimes yields directional artifacts in the processed images. On the other hand, an advantage of a causal MRF model approach is the possibility of reduced complexity on-line processing for 2-D image data. That is, in the case of sequential image transmission, received image data can be processed immediately without waiting the arrival of other data in the image space. A more important advantage of the causal MRF model approach is the availability of a variety of useful tools developed for 1-D Markov chain problems such as speech recognition [9, 10, 151]. For example, the recursive paradigm and its computationally useful algorithms can be employed for the solutions of various image processing problems by means of causal MRF modeling.