Introduction

Typical data sources have complex structure, or we say that they exhibit memory. In this chapter we study some of the basic tools for describing sources with memory, and we extend the concept of entropy from the memoryless case discussed in Chapter 1.

Initially we describe the sources in terms of vectors or patterns that occur. Since the number of messages possible under a set of constraints is often much smaller than the total number of symbol combinations, the amount of information is significantly reduced. This point of view is reflected in the notion of combinatorial entropy. In addition to the structural constraints the sources can be characterized by probability distributions, and the probabilistic definition of entropy is extended to sources with memory.

We are particularly interested in models of two-dimensional (2-D) data, and some of the methods commonly used for one-dimensional (1-D) sources can be generalized to this case. However, the analysis of 2-D fields is in general much more complex. Information theory is relevant for understanding the possibilities and limitations of many aspects of 2-D media, but many problems are either intractable or even not computable.

Finite-state sources

The source memory is described by distinguishing several states that summarize the influence of the past. We consider only the cases in which a finite number of states is sufficient.