TABLE OF CONTENTS Explored in this section is the celebrated Gibbs fields approach to image restoration and segmentation proposed by Geman and Geman [130]. It is a hidden Markov model that defines images as random intensity fields regulated by their hidden edge patterns. In what follows we have slightly reformulated the theory to fit the flow of the current chapter and book. We refer the reader to the seminal paper by Geman and Geman [130] for more details on both the theory and its stochastic computation.
As before, let ? denote the set of pixels on which an intensity image u is defined. We shall use u a to denote the intensity value at a pixel a 
?. Let E denote the set of hidden undirected edges in the graph-theoretic sense (as contrast to the edges of objects in images). That is,
For convenience, we shall also use the notation ( a, b) to denote the set { a, b}. Then ( ?, E) constitute a graph structure in combinatorial theory.
Equivalently, for each pixel a ?, define its neighborhood by
Let 
denote the neighborhood system. Then the graph structure ( ?, E) is equivalent to the neighborhood structure ( ?, 
) . The notion of neighborhood explicitly conveys the topological concept of closeness, which is beneficial for modelling spatial correlations...
