165.

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large constant, and x=0 if the number of edges occurring within four pixels is less than 3 (e.g. Figure 6.8a and b), otherwise x=1.

6.2.3 Posterior energy for image restoration

The methodology of constructing a posterior energy function as described in Sections 6.2.1 and 6.2.2 is designed for image segmentation purposes. Sometimes this approach is called a piecewise constant restoration technique. The term ‘piecewise constant’ has the same meaning as the term ‘patch’. A patch is an area in which every cell takes the same value (or symbol). Normally, remotely sensed imagery classification based on single pixels cannot give meaningful results because the patterns that we wish to map are patches, such as forested areas or agricultural fields. However, by incorporating contextual information, using prior energy, one can upgrade the pixel-based interpretation in order to produce results that are more patch-like. Such a classification process is equivalent to recovering the patches that are contaminated or disguised by noise, hence the term piecewise constant restoration.

Another approach is piecewise continuous restoration. In this case, we are not dealing with image segmentation, but are attempting to recover the true grey values in an image, or interpolating a piecewise continuous surface (i.e. the label set L becomes grey values or real numbers) in a noise fading environment. The methodology described here uses posterior energy with a piecewise smoothness assumption. That is, we use the smoothness assumption as prior information, but discontinuities can be allowed as in the line process. This assumption is especially suitable for restoring remotely sensed images containing independent identical distribution (i.i.d.) noise, since real world scenes are piecewise continuous.

Again, the first step is to construct the prior energy. For simplicity, we assume that only pairwise cliques are active. Since we are dealing with continuous values (because the label set L now contains continuous value), and noting that w_r and w_r′ both now represent grey values or real numbers, prior energy now becomes:

(6.21)

where the function g(w_r−w_r′) has to satisfy the following condition:

(6.22)

and

(6.23)

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