DILATION

1.  The so-called DILATION method of measuring the (capacity) fractal dimension (D) is one of several ways of determining D related to the length of the border of an image.  It was originally developed by Flook (1) and has been extensively used to measure the D of cellular borders (2,3).  DILATION has several features that make it superior to other methods (3).  The basic idea in the standard DILATION method is that one takes convolution kernels of different sizes (diameters) and convolves them with the image border.  Then the resultant area is divided by the diameter of the kernel and the log of that result is plotted against the log of the kernel diameter.  A fractal object gives a straight line with slope S and D = 1-S.

2.  Since the conventional (convolution) application of DILATION has not been developed for general purpose computers, a macro has been developed for use with NIH IMAGE that gives a reasonable approximation to the DILATION operation and with comparable results.  Here, the basic operation is to replace each object (e.g., black image border) pixel with a 3x3 array of (black) pixels.  The operation is continued with successive passes over the cumulative image up to some final pass.  The macro works with NIH IMAGE, version 1.57, and can be fetched using anonymous ftp from zippy.nimh.nih.gov in the /pub/nih-image/user-macros directory.

3. The procedure for use of the macro is as follows:
	a.  Open the NIH IMAGE application program.
	b  Open an image file.  The image should be a one-pixel-wide-border, binary image (black border on a white background).  (NOTE:  An interactive algorithm for obtaining such an image is detailed in the document Ògray_to_binary.txtÓ, which is also located on zippy in the documents directory.)
	c.  In NIH IMAGE, select the Load Macros under the Special Menu.  Locate the User Macros folder and open (double click) the Fractal Dilation item.  Then, with the image file high-lighted, select the Fractal Dilation item under the Special Menu.  The measurements then proceed automatically and the results are displayed on the right side of the CRT monitor.
	d.  When the measurements are completed, they can be saved as a named (e.g., FileName.ext) file.  Once saved, the result file should be closed.

4.  The output files consists of three columns:
	A.  The measuring element (equivalent to kernel diameter).
	B.  The counts of the result of each measurement with each A.
	C.  (B) counts column / (A) measuring element column.

5.  The file is stored as a tab-delimited file and can be opened into a Graphics application program, such as Kaleidagraph.  One can plot log B vs. log A and/or log C vs. log A.  Then a power (log-log) relationship can be fitted to the plot(s).  The slope of B vs. A is S(1) and the slope of C vs. A is S(2).  D = 2 - S(1) or D = 1 - S(2).  They should be the same.

6.  Preliminary tests have shown that there are several (known) limitations or problems with the above procedures.  For example, the parts of the plots in the range of small measuring-element sizes can be a problem, which leads to deviations of the plots from straight lines.  This is a consequence, in part, of the  fact that, if borders have significant straight line (Euclidian) components, the S(2) slope approaches zero and D = 1, with small measuring elements..  Also, if the adjacent borders are very close to one another, the | slopes | are too high.  The practical solution is to omit (mask) these measurements from the plot to assure a better straight line plot for the remaining data.  A second potential problem arises when the measurements with large measuring elements exceed the bounds of the CRT image frame.  These results are lost and give erroneous plots.  The practical solution is to prevent this problem is to make the image occupy, say, only the middle 1/2-2/3 of the CRT frame--either by decreasing the size of the image or choosing a larger frame.
	Finally, the method has been tested extensively, but not exhaustively, against a variety of deterministic and natural fractals for a variety of potential problems, such as anisotropy, etc. and no serious difficulties have been encountered.  However, if further field tests reveal problems, please contact the authors.  Thank you.


(1)  A. G. Flook, Powder Technology, 21:295-298, 1978; Acta. Stereol., 1:79, 1982.
(2)  T. G. Smith, et al., J. Neurosci. Methods, 27: 173-180, 1989; Neurosci., 41: 159-169, 1991; Brain Res., 634: 181, 1994.
(3)  T. G.Smith, Jr. and G. D. Lange, In: Fractal Geometry in Biological Systems: An Analytic Approach, P. M. Iannaccone (Ed.), C. R. C. Press, in press.