173.

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Page 253

A straightforward way to find the minimum of the expression shown in Equation (6.33) is to find the point where the gradient (the first derivative of g(d_i−w)) is zero:

(6.34)

Recall Equation (6.24), which gives g′(d_i−w)=2(d_i−w)·h(d_i−w). Substitute Equation (6.24) into Equation (6.34), and perform expansion, and the following expressions result:

(6.35)

It can be seen that Equation (6.35) is similar to Equations (4.15) and (4.23), which are used in Chapter 4 as part of the fuzzy c-mean and fuzzy maximum likelihood calculations. The interaction function h(d_i−w) used here is also acting as a weighting parameter, although the methods for calculating the weights may be different in the two cases. In the robust Mestimator, the function h(d_i−w) (see Equation (6.24)) is an even function, i.e. h(d_i–w)= h(w−d_i), and provides adaptive weighting behaviour, that is, h(d_i−w) should be decreasing when |d_i–w| is increasing, and h(d_i−w) should provide a higher weight if |d_i–w| is sufficiently small. This interaction concept has also been applied in the discontinuity adaptive MRF model (Li, 1995b) as noted above.

When one is defining a penalty function, the questions ‘how good is the penalty function?’ or ‘does the defined penalty function match our requirements?’ may arise. A convenient way to resolve these questions is by way of an interaction function h(η_i), where η_i=d_i−w (in Equation (6.35)), or η_i=w_r−w_r′ (in Equation (6.24)). In searching for the solution relating to the penalty function g(η_i), one generally has to resolve the first derivative of g(η_i) (e.g. Equation (6.30) and (6.34)). Coding g′(η_i) in terms of both η_i and the interaction function h(η_i), the solution is then dependent on η_i and h(η_i). One can obtain the desired output by carefully manipulating the interaction h(η_i) given η_i, and the penalty function g(η_i) can then be derived (Li, 1995b)

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