184.

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(6.48)

The second assumption is that w depends on the labels in the local neighbourhood (i.e. the image has Markovian properties). Based upon these two assumptions, and using Bayes theorem, we get:

(6.49)

The MAP-MRF estimation is now transformed to maximise Equation (6.49), which can be done by maximising P(d_r|w_r)·P(w_r|w_Nr) on each pixel r locally.

The algorithm for ICM is shown in Figure 6.18. Note that the ICM algorithm can converge to a local minima very quickly. Our experience suggests that ten iterations are sufficient to allow the procedure to converge.

6.5.2 Simulated annealing

Simulated annealing (SA) is a type of stochastic relaxation algorithm. It was first proposed by Metropolis et al. (1953) to simulate the behaviour of a system containing a large number of particles in thermal equilibrium. Geman and Geman (1984) applied a similar idea to image segmentation. The SA algorithm designed by Geman and Geman (1984), called the Gibbs sampler, generates a new label w_r′ for each pixel r based on conditional

Figure 6.18 Iterated conditional modes algorithm.

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