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larger the γ value, the wider the corresponding function shapes. Interaction h(η_i) decreases as|η_i| increases and reaches zero (i.e. h(η_i)=0) if |η_i| becomes too large.

The algorithm for the robust M-estimator is listed in Figure 6.13. Note that convergence is dependent on the initial state (although generally the process is initialised using a traditional unweighted least-squares estimate. The accuracy of the estimate also depends on the suitable choice of interaction function and interaction range parameter γ. How to overcome these issues is an important topic needing further investigation.

6.4 Parameter estimation

A probability model is not complete if the model-associated parameters are not fully specified, even if the functional form and the distribution are known. A good choice of parameters can successfully restore or segment a noise-disturbed image. Conversely, a poor selection of parameter values will usually generate poor results, as the following example demonstrates.

Two images are shown in Figure 6.14a. Each has two grey levels, 40 and 80, with pair-site parameters β={−0.6, 0.6, 0.6, 0.6} (Figure 6.14a, left-hand side) and β={1, 1, 1, 1} (Figure 6.14a, right-hand side), using the algorithm in Figure 6.4. After contamination by Gaussian noise with mean μ=0 and variance σ²=40², the images shown in Figure 6.14b are derived. The results achieved using the known parameters (i.e. the values of μ, σ², and β that were used in generating the images) and incorrect estimates of β (i.e. {−2, −2, 2, 2}) are shown in Figure 6.14c and d, respectively.

In comparison with the study of probability function formulation, the

Figure 6.13 The robust M-estimator algorithm.

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