Classification Methods for Remotely Sensed Data, Second Edition

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where a is a constant. The penalty function g(η) defined in terms of such a backward, solution-driven process can be shown to fit the user’s requirements.

Recall from Equation (6.35) that the mean derived from the robust M-estimator is controlled by the adaptive weights, i.e. by the interaction function h(η). One possible choice for h(η) can be (Tukey, 1977):

and

where γ controls the interaction range. Large |η_i| can still interact with the estimate if a larger γ value is used, while smaller γ values only enable those pixels with smaller |η_i| values to contribute. Tukey (1977) suggests that γ should be defined as γ=6·median{η_i, i} for consistency with the Gaussian distribution. The corresponding shapes of h(η) and g(η) defined in Equations (6.37) and (6.38) are illustrated in Figure 6.12. It is shown that the

Figure 6.12 Function shapes corresponding to the terms h(η/γ) and g(η/γ) in Equations (6.37) and (6.38) showing the effect of different γ values.