Measuring the Fractal Dimension of Signals
Many natural processes described by time series (e. g., noises, economical and demographic data, electric signals... etc.) are also fractals in the sense that their graph is a fractal set (Maragos and Sun 1993). Thus, modeling fractal signals is of great interest in signal processing.
Considering the importance of this index and the impact of its use in practice, the precision of its estimate is necessary. Methods of Box-counting and of Minkowski- Bouligand prove then ineffective due to the fact that they suffer from inaccuracy as we already mentioned. Inspired by the Minkowski-Bouligand method, a class of approaches to compute the fractal dimension of signal curves or one-dimensional profiles called “covering methods” is then proposed by several researchers.
These methods consist in creating multiscale covers around the signal’s graph. Indeed, each covering is formed by the union of specified structuring elements. In the method of Box-counting, the structuring element used is the square or limp, that of Minkowski-Bouligand uses the disk.
Dubuc et al. (1989 and Tricot et al. (1988) proposed a new method called “Variation method”. This one criticizes the standard methods of fractal dimension estimation namely: Box-counting and Minkowski-Bouligand. Indeed, “Variation method” applied to various fractal curves showed a high degree of accuracy and robustness.
Maragos and Sun (1993) generalized the method of Minkowski-Bouligand by proposing the “Morphological covering method” which uses multiscale morphological operations with varying structuring elements. Thus, this method unifies and improves other covering methods. Experimentally, “Morphological covering method” demonstrated a good performance, since it has experimentally been found to yield average estimation errors of about 2%-4% or less for discrete fractal signals whose fractal dimension is theoretically known (Maragos and Sun 1993). For deterministic fractal signals (these signals will be detailed further in this chapter) Maragos and Sun developed an optimization method which showed an excellent performance, since the estimation error was found between 0 % and 0.07 %.