WS Procs 975 x 65

8
This implies that the object has nearly the Euclidean shape. Fractal dimension of the
surface is significantly bigger than the Euclidean dimension of plane E = 2 and has
therefore evidently fractal nature. The fractal measure of the interface determines the size
of the fractal structure surface (in pixels) and is KBW/Kmax = 362.79 % of the image’s
planar surface, where Kmax = (512 × 512) pixels. Fractal measure of the structure volume
is KBBW/Kmax = 40.97 % and its surroundings KWBW/Kmax = 59.03 %, where
Kmax = (512 × 512 × 255) voxels (volume pixels). The fractal measure of the interface
tells us therefore that the surface of the fractal structure is more than three times larger
than the surface of plane.
6 Conclusion
It is worth mentioning that the fractal dimension of the deterministic fractal structures
(e.g. Sierpinsky carpet) is immutable in the whole range of sizes of the net applied on the
image using box counting method (or level of filter using wavelet – Haar –
transformation) and is the same as the theoretical fractal dimension value. Similar fractal
dimension values were also obtained using other types of integral, e.g. the Fourier
transformations. In this case the values of fractal parameters are influenced by broader
vicinity (step function is replaced by harmonic function).
The fractal dimension evaluated using the Haar transformation offers in contrast to
the classic box counting method a much wider range of usability. The method can also be
used to determine the fractal dimension of colour structures (e.g. greyscale or full colour
images). This also allows determining the fractal dimensions of surfaces specified by
shades of grey or colour components. This can be also used to determine the fractal
parameters of surfaces or three-dimensional structures like distribution of the mass in the
space or electrical potential in space. The theory of the fractal structures in Edimensional
Euclidean space is described more closely e.g. in [4, 5].
Software for fractal analysis of the images is provided free on the Internet
http://www.fch.vutbr.cz/lectures/imagesci.
Literature
1 . B. B. Mandelbrot, Fractal geometry of nature. New York: W.H. Freeman and Co.
(1983)
2 . M. J. Barnsley, Fractals everywhere, New York:Academic Press Inc. (1993)
3 . G. Strang, Wavelets Transforms versus Fourier Transforms, Bulletin of the
American Mathematical Society, 28, 288 (1993)
4 . O. Zmeskal, M. Nezadal, M. Buchnicek, Chaos, Solitons & Fractals, 17, 113 (2003)
5 . O. Zmeskal, M. Nezadal, M. Buchnicek, Chaos, Solitons & Fractals, 19, 1013 (2004)