ABSTRACT
Figure 7.2a shows another common exactly self-similar fractal, the triadic Cantor set. The construction method is shown in the plot where, at each step in the generation of the fractal set, the middle third is removed from the remaining line segments. The iteration process begins on the unit line and proceeds ad infinitum to construct the set. Figure 7.2b shows a
FIGURE 7.1 Exactly self-similar and statistically self-similar fractals: (a) The Sierpinski gasket: A fractal object which is exactly self-similar over all scales. (Each of the circles contains selfsimilar parts of the whole gasket at different scales.) (b) The two-dimensional trajectory of ordinary Brownian motion (The right-hand plot contains the first 1/16th of the trajectory of the left-hand plot blow up to maintain the same degree of resolution between each plot.) (From Addison, P. S., Fractals and Chaos: An Illustrated Course, CRC Press, Bristol, 1997.)
3-D plot of a Mexican hat wavelet transform for the triadic Cantor set (Arneodo et al., 1989). The branching structure of the set is easily seen in the transform plot. Figure 7.3 contains a regular snowflake fractal generated in the plane together with transform plots at three a scales using the radial (2-D) Mexican hat wavelet (Argoul et al., 1989). Contour lines, set at an arbitrary value, show the construction rule of the fractal snowflake in Figure 7.4. As the scale of the wavelet tends to zero, the transform plot approximates more and more the
FIGURE 7.2 Wavelet analysis of the triadic Cantor set: (a) Construction of the triadic Cantor set. (From Addison, P. S., Fractals and Chaos: An Illustrated Course, CRC Press, Bristol, 1997.) (b) Transform plots for the Triadic Cantor Set: Note that ( ( ). ( , ) )/sgn T T a b 1 2 is plotted against ln(a) and b. (From Arneodo, A. et al., Wavelets, 182-196. Springer, Berlin, 1989.)
FIGURE 7.3 The wavelet transform of a regular snowflake. The snowflake is shown in (a). The scale parameter a is successively divided by the same factor l = 3: a = a* (b), a = a*/3 (c), a = a*/32 (d). T(a,b) is expressed in alog(5)/log(3) units in order to reveal the self-similarity of the geometry of the snowflakes. (From Argoul, F. et al., Physics Letters A, 135(6), 327-336, 1989.)
FIGURE 7.4 Isocontour lines of the wavelet transform of the snowflake fractal. The isocontour line T(a,b)/alog(5)/log(3) = k (k arbitrary chosen) for different values of the scale parameter: a = a* (a) a = a*/3 (b), a = a*/32 (c), a=a*/33 (d). (From Argoul, F. et al., Physics Letters A, 135(6), 327-336, 1989.)
snowflake itself. See also Antoine et al. (1997), who analyzed a fractal Koch curve in their paper concerning the characterization of shapes using the information gained from the maxima lines (both modulus maxima and ridges) of the continuous wavelet transform. The Sierpinski gasket, Cantor set, snowflake fractal and the Koch curve are exactly selfsimilar. However, very few examples of exact self-similarity are to be found in nature, (e.g. some fern shapes exhibit nearly exact self-similarity over a few scales). Most natural fractals exhibit stochastic self-similarity and consequently most research concerning fractal objects and processes in nature focuses on stochastic fractals.
