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The classification resulting from the use of hierarchical fuzzy rules (partitions from 5 to 20 on each dimension, i.e. a total 5²+ 6²+…+20² rules) is shown in Figure 4.15d. The resulting classification decision boundary is smoother than that shown in Figure 4.15b and c. This interesting property of the hierarchical fuzzy rule methodology is considered further below.

The hierarchical fuzzy rule approach is used to overcome the membership size selection problem for fuzzy partitions. As the fuzzy partitions become smaller, and if there are insufficient training patterns, the result is the presence of dummy subspaces (i.e. subspaces for which no fuzzy rules are available). Pixels falling within these dummy subspaces cannot be classified. This difficulty was mentioned earlier. An example is illustrated in Figure 4.15e in which the number of training patterns is fifty (selected at random) and the number of fuzzy partitions in each dimension is set to twenty. With such fine fuzzy partitions and with a small number of training patterns then dummy subspaces (shown in black) tend to occur. The image generated by the hierarchical fuzzy rules procedure (using five to twenty partitions on each dimension) is shown in Figure 4.15f. A comparison of Figures 4.15f and d shows the influence of training sample size.

The combination of hierarchical fuzzy partitions results in smoother decision boundaries, as illustrated in Figure 4.16. The idea can be explained using two different sizes of fuzzy partition, A and B (Figure 4.16a). The decision boundaries corresponding to A and B are shown as dotted lines. If the classification process is carried out in terms of combining A and B then several possibilities for forming decision boundary may result. Two of those possible situations are shown in Figure 4.16b and c. Figure 4.16b illustrates the decision boundary determined by partition size A, which exhibits considerable blockiness. Figure 4.16b illustrates the decision boundary formed by the combination of partition size B and part of partition A. In this case, the decision boundary is less blocky. If more, different, fuzzy partitions are employed in a hierarchical fuzzy rule classification mechanism, the decision boundaries will be smoother than those resulting from the use of a single fuzzy partition.

4.6 Fuzzy classification: interpretation of mixed pixels

Most remotely sensed image classification procedures operate under the hypothesis that each pixel is perfectly pure, i.e. the ground area represented by a pixel is occupied by a single information class. If the pixel resolution is coarse in comparison with the variability of land cover objects then it is likely that areas representative of more than one information class will be contained within a single pixel. Such pixels are called mixed pixels. The determination of the multiple class membership of individual pixels can be regarded as the removal of classification ambiguities (e.g. Mathieu-Marni

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