135.

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to use these three parameters to describe the texture pattern. An example is shown in Figure 5.18, in which images (a), (b) and (c) are constructed from Figure 5.10b by using the mean of entries of vector θ (with neighbourhood N defined as {(0, 1), and (−1, −1)}), variance σ_u², and mean value δ_y, respectively. The practical use of these methods is illustrated in Figure 5.19.

5.5 The semivariogram and window size determination

Since textural features are scale-dependent, the choice of window size has to be made carefully. If the window size is too small relative to the texture structure, the extracted texture feature will not accurately reflect the real texture property. In order to estimate the optimal window size, the semivariogram can be used (Woodcock and Strahler, 1983; Woodcock et al., 1988a, b; Curran, 1988; Curran and Atkinson, 1998). The use of the semivariogram in texture estimation is described in below.

Given a transact across an image, where the digital number z of each pixel i, denoted by z(i), is extracted at a predefined interval or lag h, then the relationship between pairs of pixels z(i) and z(i+h) can be expressed as the average variance of the differences between all such pairs. As the per-pixel variance is half this value, the semivariance S² for pixels at distance h apart is defined by:

(5.52)

The average semivariance for lag h is:

(5.53)

S²(h) is an unbiased estimate of the average semivariance γ(h) in the population and can be used as a measure of dissimilarity between spatially separate pixels (Woodcock and Strahler, 1983; Woodcock et al., 1988a, b).

In general, the semivariogram has a shape similar to that shown in Figure 5.20. The choice of a suitable window size h′ should be made according to the criterion that, at lag h′, the semivariance has stopped increasing, or the rate of increase of the semivariance has fallen significantly. A suitable window size, h′, is indicated in Figure 5.20. We have computed the semivariogram for the test images (Figure 5.21) using a horizontal transect, and a window size of 16 was determined to be the most appropriate.

Atkinson and Lewis (2000) consider the use of the semivariogram for the estimation of image texture, and report that Carr and Miranda (1998) found that the use of the variogram produced classifications with greater accuracy than those based on the grey level co-occurrence matrix (GLCM).

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