## Fractal Dimension

The recent theory of fractals introduced by Mandelbrot [53] provided a new framework for analyzing complex geometric shapes, particularly their roughness, which can be quantified with the fractal dimension. Consider a geometric object that resides in an N-dimensional space where N is the smallest integer that allows the space to contain the object, such as N = 2 for a curve, or N = 3 for an arbitrary surface. The number of small spheres (or cubes) with diameter (or side) e needed to cover the object is

where a is a scaling constant and D is the Hausdorff dimension. A line or a smooth curve have D = 1, indicating that these are topologically 1D objects embedded in a 2D Euclidean space and D = N â€” 1. However, in the same 2D space, the Hausdorff dimensions of meandrous curves with numerous turns are not integers and can have, for example, values as high as 1.98 for very rough curves. In these cases where the Hausdorff dimension is fractional, it is also called fractal dimension. The same observations apply to surfaces in 3D space where D is 2 for smooth surfaces and increases toward 3 with increasing roughness. Fractal dimension can be used as an image texture metric if the image structure is considered as a surface in a 3D space where two dimensions are those of the image plane and the third is the pixel intensity. The area A(e) of this intensity surface can be expressed as

as a function of the element e which can take sizes such as one, two, or more pixels for a digital image. The fractal dimension D of the image can be calculated by using the area A(e) estimated at several sizes of e and applying linear regression on log A(e) = log a +(2 â€” D) log e. (56)

The value of A(e) can be estimated with the box-counting concept [54] which has been applied to medical image analysis [55], as discussed and illustrated in Chapter 21 in the context of mammogram texture quantification.

The fractal dimension also can be computed by representing the image intensity surface with a fractional Brownian-motion model. According to this model, the distance Ar between pairs of pixels (x1; y1) and (x^ y2) given by

and the absolute difference between the intensity values I (x1; y1) and I (x2, y2) of these pixels

are related with

where E[.] is the expectation operator, b is a proportionality constant, and H is the Hurst coefficient. The fractal dimension [56,57] is given by

In practice, the smallest value of Ar is Armin â€” 1 pixel and the highest value Armax is dictated by the size of the structure under consideration. To limit the values of Ar further, only integer values can be used [56], in which case Ar may be computed along horizontal and vertical directions only. The value of H is computed with linear regression on log E[AlAr]â€”log b + H log Ar (61)

using the selected Ar values. Rough textures yield large values for D and low values for H.

Fractal dimension has been used for texture analysis in ultrasonic liver images [56,58], radiographic images of the calcaneus [59], mammograms [55], colorectal polyps [60], trabecular bone [61], and CT images of pulmonary parenchyma [62]. Computation of fractal dimension using maximum likelihood [59] and fractal interpolation functions [63] also have been suggested.