Many natural objects have irregular, seemingly complex shapes. Yet often those complex shapes are assembled from simple structures. A good example is the fern leaf shown in Figure 10.1. This seemingly complex structure is made from small leaflets arranged left and right along a small stem. Many of these stems grow up and down from a larger, somewhat horizontally oriented stem. These horizontal stems, in turn, grow right and left from the main stem. At each level, the structure is similar to the whole: One of the branches growing sideways off the main stem, if turned by 90° and magnified, would look like the whole fern leaf, a phenomenon called self-similarity. In the human body, a good example are the lungs, where the descending trachea branches into the bronchial trees of the left and right lungs. Each bronchial tree consists of repeatedly (iteratively) smaller bronchial tubes branching off larger ones. The arterial and venous blood vessels, in the liver or kidney follow a similar rule: A major supply blood vessel splits into smaller vessels, which in turn split into even smaller vessels down to the level of the capillaries. The concept of self-similarity is widespread in living systems. In many cases, as in the example of the fern, a set of rules can be formulated that generates the complex object from its primitive structural elements. As explained later in this topic, this set of rules is associated with a metric, the fractal dimension, which can be interpreted intuitively as a metric of the complexity of the shape. Shapes and textures occurring in medical images can be classified by using methods that estimate the fractal dimension, and since the early 1990s, the use of fractal approaches has become a standard method in quantitative image analysis. Despite its popularity, the use of the fractal dimension to describe complexity in medical images depends on various factors. Depending on the image modality, image processing steps, and the method to estimate the fractal dimension, highly diverging results may be obtained. A thorough understanding of mathematical fractals, the property of self-similarity, and image operators to estimate fractal properties are needed.
FIGURE 10.1 A seemingly complex structure such as a fern leaf is a complex arrangement of simple structures.
