FRACTAL GEOMETRY
Fractal geometry is a generalization of conventional (Euclidean) geometry which allows for the concept of a non-integral dimension, i.e., (units) ^D, where D may not be an integer, but a fraction, and hence the name ÒfractalÓ. D is called the Òfractal dimensionÓ and is the conventional measure of fractal objects (often images). If one is interested in globally describing the shapes of objects quantitatively, one can associate D values with complexity of form. Analytic Euclidean geometry does not easily lend itself to this goal, but fractal geometry does. This is largely because complexity and scaling are intimately related. For example, to the microscopist the feature of an object that continues to reveal more detail as it is magnified is its increase in morphologically complexity (e.g., increased resolution). It is precisely this property of fractal objects, namely, self-similarity or scale invariance, where an object appears qualitatively the same, irrespective of magnification, that suggests that fractal geometry might be a good model to provide measures of complexity. Indeed, the fractal dimension measures the rate of addition of structural detail with increasing magnification. Fractal dimension (D), therefore serves as a quantifier of complexity, with an increase in D representing an increase in complexity.
It should be emphasized that D is strictly a descriptive measure; it a statistic, like a mean and does not necessarily imply any underlying mechanism of form generation. In general, connections between the empirical values of D and any specific, say, growth, mechanisms require the answering of specific experimental questions and not statistical or mathematical ones. One can, for example, view D in much the same way that physical optics might view refractive index.. That is, as a measure of a property of some object or material (even though in the case of this measure a good deal is known about the underlying mechanisms leading to its value).
In general, there are two basic approaches to measuring the fractal dimension of an object. The first, and most commonly used, basically measures the lengths or distances between points on the border of, say, a binary image (i.e, a one-pixel-wide black border on a white background). The resultant D is called the capacity dimension. The second method counts border pixels located within discs of various diameter, where the discs are randomly centered on the border. The resultant D is called the mass dimension. When a distance measure is applied, the side of a pixel becomes the unit of length. When a pixel counting measure is applied, the border pixel becomes the unit of mass.
It should be noted that, with these length-related methods the magnitude of the resultant measure (perimeters, counts/diameter, etc.) increases as the measuring element decreases in size. In a deterministic or true fractal, like Koch figures ,this increase continues without limit. This is an illustration of the important and defining property of fractals, viz. self-similarity or scaling symmetry or invariance. Such universal self-similarity exist only as mathematical formulas or in computers. Real world or natural fractals are only self-similar in a statistical sense and have straight lines in log (resultant measure) vs.log (measuring element) plots over a limited extent. The final D value calculated is an average property of the whole object and has no spatial locality. All of the methods employed to calculate D are essentially plots of log length vs. log length.
All of these methods consistently underestimate the values of deterministic fractals by a few percent. This is a consequence of the fact that a finite, digitized image, with a limited number of pixels, cannot realize the detail implicit in a deterministic fractal. This error is probably found in the measurements of natural fractals as well, but since it is a consistent, and not random, one and since most results are used comparatively, the error probably does not significantly affect the conclusions drawn
The characteristics of cellular morphology that most influence the magnitude of D are the profuseness of branching and the ruggedness or roughness of the border, with increases in either leading to a larger D. This means, of course, that two cells that look very different (e.g, one with a smooth border and many branches; the other with few branches and a rugged border) may have the same D. This result emphasizes that, with such global measures, D provides no unique morphological specification. As an aside one might note that the concepts of Òborder roughnessÓ and Òprofuse branchingÓ relate to a given image at a particular magnification--at a higher magnification a rough border might appear as diffuse branching and, at a lower magnification, profuse branching might look like rough border. This is another manifestation of self-similarity.
The exact method employed to measure D does not depend on whether an image is self-similar. That is determined by the log-log plots. The range of the linear slope of those plots identifies the range of self-similarity. In the matters considered here, it is only the border that is fractal. The structureless interior is Euclidian, not fractal. One can show this by doing an analysis of a filled silhouette versus a border of a Koch figure with known D. Only the border image gives the correct result.
Most of the experiments reported in the bio-medical literature that apply the concepts of fractal geometry to cellular morphology involve the use of length-related measures of D. This is also true to those methods available at present (3/95) for NIH IMAGE. This reflects the historical precedence of these over mass measures.
Experience has demonstrated that fractal geometry has proven a useful tool in quantifying the structure a wide range of idealized and naturally occurring objects. The range of application extends from pure and applied mathematics,through physics and chemistry to biology and medicine.
RELATED TOPICS AVAILABLE BY FTP FROM ZIPPY.NIMH.NIH.GOV:
ImageFractal, in the nih-image_spinoffs directory
fractal_dilation.txt, in the documents directory
gray_to_binary.txt, in the documents directory