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An example of image restoration using a remotely sensed image is presented in Figure 6.11. In this example, an image was contaminated by zero mean, white Gaussian noise with a standard deviation of 80. Three values for the weighting parameter q were chosen as 0.5, 2 and 10, respectively, to perform restoration using a quadratic function as shown in Equation (6.25) as the penalty function, and a gradient descent step of k= 0.01. It is clear that higher values of q generate smoother results.

Note that a more satisfactory restoration might be achieved by choosing an adaptive penalty function, for example the discontinuity adaptive (DA) MRF model described by Li (1995b), which is constructed on the basis of the Lagrange-Euler equation (Courant and Hibert, 1953), and design h(a −b) in Equation (6.24) as an adaptive function (one kind of adaptive functions h(a−b) is discussed in Section 6.3). Li (1995b) provides details of this method.

6.3 Robust M estimator

The aim of robust estimators is to determine effective estimates in the presence of noise. This noise may be the result of the mixing-in of values that do not belong to the population being analysed. Statistical methods of classification are generally based on the mean vector and variance-covariance matrix of each class. The results achieved by a statistical classifier will therefore be influenced by the accuracy with which the class mean and variance-covariance matrix are estimated. If the training data for a given class are contaminated by pixels that do not belong to that class, or by erroneous values, then the traditional least-squares method will generate misleading results. The use of robust methods of estimation can detect (and down-weight) outliers in order to provide better estimates of the statistical parameters required.

Several types of robust estimators have been proposed in the literature, for example, (1) the rank transformation (R-estimator) (Huber, 1981); (2) the repeated median (RM-estimator) (Siegel, 1982); and (3) the maximum likelihood criteria (M-estimator) (e.g. Tukey, 1977; Jhung and Swain, 1996). In this section, the M-estimator for mean estimate is considered. For variance estimates, the reader may refer to Jhung and Swain (1996).

Let w denote the mean value being estimated from m observations (e.g. pixels) d_i, where 1≤i≤m. The residual error η_i at observation d_i can be expressed by η_i= d_i−w, and the error penalty function is g(d_i−w). Note that the distribution of error η_i is not actually known; the only information is that the η_i have an independent identical distribution (i.i.d.). The robust M-estimate is derived from minimising the sum of penalty functions:

(6.33)

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