## Fractal Analysis

"Fractal geometry," proposed and initially developed by Mandelbrot [28], can provide a mathematical description for complex shapes that are not easily described by Euclidean geometry. "Fractal objects" often possess an invariance or "statistical self-similarity" when observed under different scales. This property can be used to determine a "fractal dimension" that quantifies the complexity of the object. Fractal dimension is consistent with the concept of spatial dimension, but is not restricted to being an integer. For example, a higher fractal dimension represents a more complex or space-filling object [28-30].

The fractal concept has been applied to many fields in medicine [31], and in particular has found widespread application in the description of radiographic images [21, 3236]. In such an application, the fractal dimension has been calculated to characterize the inherent texture in regions of the image. Caldwell etal. [21] found that a fractal dimension could be calculated for mammographic parenchyma, which was related to the qualitative categories of the Wolfe grades. In fact, Wolfe's [2,3] original description of regions of fibroglandular tissue as sheetlike areas of density was suggestive of textural analysis for mammographic parenchyma. As seen in the images and corresponding surface plots from Fig. 5, a breast with a

FIGURE 5 Wire-frame representations of two digitized mammograms. The gray level of each pixel determines the height of the terrain. (Upper) Low mammographic density. (Lower) Extensive density. Reprinted with permission from Byng JW, Boyd NF, Fishell E, et al. Automated analysis of mammographic densities. Physics in Medicine and Biology 1996; 41: 909-923.

FIGURE 5 Wire-frame representations of two digitized mammograms. The gray level of each pixel determines the height of the terrain. (Upper) Low mammographic density. (Lower) Extensive density. Reprinted with permission from Byng JW, Boyd NF, Fishell E, et al. Automated analysis of mammographic densities. Physics in Medicine and Biology 1996; 41: 909-923.

small degree of density will have an image with coarse texture (upper part of figure), due to good contrast between the connective tissue and the predominately fatty glandular tissue. Similarly, when there is a high degree of mammographic density, the image will appear smoother, which should be reflected by a lower fractal dimension.

One method for calculating the fractal dimension follows the approach of Lundahl et al. [32]. This calculation is analogous to a "box-counting technique" described by Barnsley [29]. In this approach, the brightness i of each pixel at (x y) in a digitized image can be treated as a vertical dimension accompanying the two-dimensional assignment of pixels (of size e x e) in the digitization process. This creates a surface shown schematically in Fig. 6a.

The first step of the measurement technique is to determine the area, A(e), of such a surface for a given pixel size, e. This surface area is calculated as a sum of the area of each pixel, e2, plus the contributions of the "exposed" sides of the boxes (difference in height (or pixel value) between neighboring pixels):

FIGURE 6 (a) Calculation of A(e) in the measurement of fractal dimension, (b) Illustration of the regression of log [A(e)] vs log [e] in the measurement of fractal dimension. Reprinted with permission from Byng JW, Boyd NF, Fishell E, et al. Automated analysis of mammographie densities. Physics in Medicine and Biology 1996; 41: 909-923.

FIGURE 6 (a) Calculation of A(e) in the measurement of fractal dimension, (b) Illustration of the regression of log [A(e)] vs log [e] in the measurement of fractal dimension. Reprinted with permission from Byng JW, Boyd NF, Fishell E, et al. Automated analysis of mammographie densities. Physics in Medicine and Biology 1996; 41: 909-923.

a(£) = E e2 + E £(iie(* ' y) - %(*> y+i)i x, y x, y

Mandelbrot [28] showed that for certain structures or images, called "fractal," there is a power law relationship between A(e) and e, the exponent being related to the "fractal dimension." Specifically, for a two-dimensional image, the fractal dimension is given by

In practice A(e) is calculated from Eq. (8), for various pixel sizes, e. A set of different effective pixel sizes can be synthesized by averaging adjacent pixels together [21]. Combinations of 1 x 1, 2 x 2, 3 x 3, and 4 x 4 pixels, etc., can be used to obtain values of e for the regression. The application of the fractal model holds if, over the range of pixel sizes, this regression is linear.

Equation (9) indicates that for an image, the fractal dimension can be determined from the slope of the regression between log [A(e)] and log [e] over a range of pixel sizes. This is illustrated in Fig. 6b. For the 60 images that we studied, the measured fractal dimension ranged between 2.23 and 2.54, similar to the range observed by Caldwell et al. [21]. In the determination of the fractal measure, the coefficient of regression between logarithms of ruler size, e, and measured surface area, A(e), over the 60 images ranged between 0.962 and 0.999, indicating excellent linearity of this relationship.

When the fractal parameter was compared on the same 60 images to the density categories of the SCC assessed by a radiologist, a correlation coefficient Rs = — 0.76 was achieved, indicating a strong negative trend (i.e., as the proportion of mammographic density increases, the fractal dimension decreases).

Taylor et al. [23] also considered the fractal dimension and regional skewness measurements as features for characterizing breast composition and found that they successfully provide a two-category discrimination ofdense from fatty mammograms.

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