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The image containing S cells (i.e. S=x·y·z), and the expected number of boxes with side L needed to cover the whole image is:
(5.7) |
By counting the total number of boxes needed to cover the image with several different side lengths L, the fractal dimension D is estimated from Equation (5.5) as:
(5.8) |
That is, D is the negative of the slope of the least squares line relating ln(L) and—ln(NL.). Since S is a constant, it will not affect the slope of the regression line and thus can be dropped from Equation (5.7) to give:
(5.9) |
The value NL is proportional to L−D and can be used as surrogate for D. The algorithm for calculating the fractal dimension using the box counting method is shown in Figure 5.5, where Lmax should not be larger than the image x and y dimensions nor larger than the range of grey levels in the
Figure 5.5 Fractal dimension estimation based on Keller’s method.
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