Measuring the Fractal Dimension of Signals: Morphological Covers and Iterative Optimization
Fractals can model many classes of time-series data. The fractal dimension is an important characteristic of fractals that contains information about their geometrical structure at multiple scales. The covering methods are a class of efficient approaches to measure the fractal dimension of an arbitrary fractal signal by creating multiscale covers around the signal's graph. In this paper we develop a general method that uses multiscale morphological operations with varying structuring elements to unify and extend the theory and digital implementations of covering methods. It is theoretically estab- lished that, for the fractal dimension computation, covering one-dimensional signals with planar sets is equivalent to mor- phologically transforming the signal by one-dimensional func- tions, which reduces the computational complexity from quad- ratic in the signal's length to linear. Then a morphological covering algorithm is developed and applied to discrete-time signals synthesized from Weierstrass functions, fractal inter- polation functions, and fractional Brownian motion. Further, for deterministic parametric fractals depending on a single pa- rameter related to their dimension, we develop an optimization method that starts from an initial estimate and iteratively con- verges to the true fractal dimension by searching in the param- eter space and minimizing a distance between the original sig- nal and all such signals from the same class. Experimental results are also provided to demonstrate the good performance of the developed methods.