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where is the number of pixels with label 1, 7112 is the number of { g , g2} neighbor pairs etc, so we are assuming that only pairs of sites are neighbors. By consideration of realizations which differ only a t pixel it can be shown that
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which is a similar model to (7.4), differing in that (7.4) allows a separate for each pixel. We can also consider a set of 1 ordered values such a grey-level image with intensity values of 0 (black) to 255 (white). This gives an auto-binomial model with
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In the case where we have two classes, labeled 0 and 1, the multi-color and the ordered M R F reduce t o the same model, the auto-logistic or Ising spin glass model with
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and (7.11) It is also possible to construct models where the class label is a continuous variable, such as the auto-normal model of Besag (1986). Geman and Geman (1984) comment that (7.9) and (7.8) allow the M R F to be specified in terms of the potential rather than in terms of the local conditional distributions, which is difficult to do in such a way that the conditions of result (7.7) are met. However, i t is not clear that the M R F model is more than locally and approximately correct in image restoration problems. Besag (1986) warns that while (7.2) mimics the local characteristics of the scene, it may be that the global properties of the assumed M R F will dominate the restoration of the image, producing long-scale dependencies between pixels and a poor reconstruction. In addition, Besag questions whether replacing a local conditional distribution based on exp{pN(g, with one based on P N ( g , will lead to a poorer reconstruction, although it will lead t o a violation of the consistency of the M R F .
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The Hopfield network is closely associated with the insights of the Hebbian learning rule introduced by Hebb (1949). This is the idea that when . . . cell A is near enough to excite cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic process takes part in one or both cells such that A s efficiency one of the cells firing B is increased (Hebb quoted in Serra and Zanarini, 1990). A Hopfield network consists of interconnected units, each of which is a binary unit is The logic device with the two states 0 and 1, say. The state of the connections between the units, which we describe by a set of weights are real valued and are modified by a learning rule. Each unit receives input from the other N units so that the input t o the unit at a fixed time is = the weighted sum of the states of the set of units. This input is compared with a threshold 0 and the following action is taken:
X , ( t 1) = 1 X , ( t + 1) = 0
if if if
>0 <0
@ { X , ( t ) }= 0
If we introduce a probability distribution on @ { X , ( t ) } ,as in the Boltzman machine, then we have a close relationship with the M R F models. For a Boltzman
machine, 2.2, p. 11)
is a Bernoulli random variable with a logistic distribution (equation
= = logistic[@(X,(t)}]. If we define neighborhood cliques so that wzl = 0 if and j are not neighbors, this model corresponds t o the MRF model (7.11). The Hopfield network has been widely explored for optimization problems, such the traveling salesperson. For further details on it and the Boltzmann machine see Pa0 (1989), Serra and Zanarini (1990) or Lippmann (1987) for a brief introduction.