By John C. Russ (auth.)
The writer integrates discussions of fractal geometry, floor modeling ideas, and functions to genuine global difficulties to supply a entire, available assessment of the sector. His paintings will equip researchers with the fundamental instruments for size and interpretation of information, stimulating extra paintings on those difficulties and, possibly, resulting in an figuring out of the explanations that Nature has followed this geometry to form a lot of our world.
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Extra info for Fractal Surfaces
1989). Another fast way to implement the Minkowski measurement for vertical elevation profiles is to perform the one-dimensional equivalent of the Euclidean Distance Map, producing a grey scale array which can be counted to produce the log-log plot. This is illustrated in Figures 27 and 28. All ofthese approaches give identieal results, differing only in the speed of implementation. ), it is necessary to fit a straight line to the data to determine the slope and hence the fractal dimension. Since the points are not uniformly spaced, and the regression fit should not necessarily weight a11 of the points the same, this is not an entirely trivial operation (Draper and Smith 1981).
The results of progressively thresholding the EDM image of feature A in Figure 22 at different pixel values, to obtain the pixels within varying distances of the boundary. These are Minkowski sausages. If the important scaling dimension is along the horizontal direction, then a structuring element consisting of a line segment parallel to the horizontal axis can be used. Aseries of line segments of different lengths are moved continuously along the profile and the area swept out plotted as a function of line length (using log-log axes).
301, in good agreement with the value of a: used in generating the line. 5t-----------i-------------~k 7 . 4 i--------------'--------------------~\o(b 7 . 5 3 so Chapter 2: Measuring the Fractal Dimension of Boundary Lines If only short profiles are available for measurement, for instance in measuring the profiles of many small particles, the usual approach is to measure each and determine an average. There is DO reason to expect this to give the "right" answer. If the profile contains regions with different local dimensions, or has a scaling dimension which is not constant, the log-log plot may npt be straight.