Classification Methods for Remotely Sensed Data, Second Edition

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Located to the right of the ‘intrusives’ area is a region underlain by granitic rocks, termed ‘granitic batholith’. This area also appears smooth but the structure is different from that of the ‘intrusives’ group of rocks. The area around the middle of the image is underlain by ‘volcano-sedimentary’ rocks, and shows a very rough texture pattern. This image is used as a basis for generating texture images in terms of fractal and other algorithms introduced in following paragraphs. The fractal texture image derived from Figure 5.10b using the algorithm illustrated in Figure 5.7 with subimage size defined as 15×15 is shown in Figure 5.10c. Brighter areas denote higher fractal dimensions, indicating that the corresponding area is texturally rougher.

5.1.3 Multifractal dimension and the function D(q)

It was noted in Section 5.1.2 that the estimation of fractal dimension takes the contributions of all windows of an image and reflects the combined behaviour of all windows. However, it is known that real-world images are not actually fractal. It is also the case that images with different texture patterns may have the same fractal dimension. Consequently, using fractal dimension to estimate the roughness of the image for segmentation purposes without regard to local characteristics may generate some confusion (Dubuc et al., 1989). To overcome this drawback, the concept of multifractals can be employed (Parisi and Frish, 1985). Multifractal means that a set’s fractal dimension differs from its subset’s fractal dimension.

Suppose one has a box of unit length (Figure 5.11), which contains a set of P uniformly spaced points. The box is divided into two equal-length sub-boxes. Each segment contains the proportions p₀ and p₁ of the P points, respectively. That is, , and p₀+p₁=1. For instance, if the number of points contained by the two sub-boxes is s and (P−s), respectively, then the ratio p₀ will be equal to s/P, and p₁ will be equal to (P−s)/P. At the next step, a recursive process is started and, at each iteration, all of the sub-boxes are divided into two equal length sub-boxes, and the percentage of points contained by each sub-box is calculated by multiplying by p₀ and p₁. In other words, at each step for each sub-box i, the process will generate two equal length sub-boxes, one containing p₀ and the other containing p₁ per cent of the points contained by their mother sub-box i. This is known as a binary multiplicative process. Figure 5.11 shows this procedure with p₀=, and p₁=.

At iteration n, a set M characterising all point distributions (in percentage terms) can be expressed by:

where