99.

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rather than of a point. If the area over which texture is measured contains two or more different regions or categories then texture measures may not prove to be useful.

The operational definition of texture features is difficult. The main texture recognition approaches can be categorised into four groups in terms of their different theoretical backgrounds. The first approach defines texture features that are derived from the Fourier power spectrum of the image under study via frequency domain filtering. Since different textures demonstrate different frequency patterns, it is reasonable to postulate that texture features are related to the distribution of spatial frequency components. The second approach is based on statistics that measure local properties that are thought to be related to texture, for example, the local mean or standard deviation. The third approach is the use of the joint grey level probability density (Haralick et al., 1973). The final approach is based on modelling the image using assumptions such as that the image being processed possesses fractal properties (Mandelbrot, 1977, 1982), or can be modelled using a random field model such as the multiplicative autoregressive random (MAR) field (Frankot and Chellapa, 1987).

This chapter presents a basic description of texture measures for image segmentation. Seven approaches based on different theoretical background are introduced. The first four methods are derived from multifractal theory; the fifth method is based on frequency domain filtering; the sixth method uses the grey level co-occurrence matrix (GLCM), and the seventh texture quantisation approach is based on the multiplicative autoregressive random field model.

5.1 Fractal and multifractal dimensions

If an object possesses the ‘fractal’ property then it can be thought of as being constructed of an infinite number of copies of itself, varying in scale from large to small. Two examples of fractal patterns are shown in Figure 5.1. The fractal dimension of an object is a measure of its complexity. The higher the fractal dimension, the more complex is the object shape or structure. For example, Figure 5.1a shows objects that have a fractal dimension of 1.58, while Figure 5.1b shows objects with a fractal dimension of 1.89. Fractal dimension can thus be applied as a measurement for texture quantisation, as complex surfaces (such as the intensity surface represented by an image) have a rough texture, while simple (smooth) surfaces have a fine texture.

Real-world images (or objects) are not truly fractal; hence, the multifractal concept was introduced. The image is considered to be composed of a finite number of subsets, and its multifractal dimension considers each subset’s property and is thus considered to be a more powerful measure of image texture (Tso, 1997). An introduction to the use of fractal and multi

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